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PSLE

<p>The figure below shows a container that is a trapezoidal prism. The container is empty at first, and then is filled with water by a pipe with diameter <strong>10 cm</strong> at a constant speed of <strong>6 m/s</strong>.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767771591862-2iqlco8wksk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzE1OTE4NjItMmlxbGNvOHdrc2sucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY4NCwiZXhwIjoxNzkyNjYyOTg0fQ.Ii5F5OxgpcUaEVxNonxW9CgO_HMMxcKTb2weKCNboS0" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find volume of water filled into the container per second in litres/s, leaving your answer in terms of $\pi$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence, find the total time needed for the container to be completely filled, correcting your answer to <strong>2</strong> decimal places.</div></li></ol>
<p>After $t$ seconds, the depth of the water in the container is <strong>1.4 m</strong> and the breadth of the water surface in the container is <strong>2.3 m</strong>.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the value of $t$, correcting your answer to <strong>2</strong> decimal places.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the volume of empty space in the container at this moment.</div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Find the area of the water surface at this moment.</div></li></ol>
<p>Correction to the retained source diagram: at depth 1.4 m, the width is $1.6+(2.4-1.6)(1.4/1.6)=2.3\text{ m}$, not the printed 2.1 m.</p>

The figure below shows a container that is a trapezoidal prism. The container is empty at first, and then is filled with water by a pipe with diameter **10 cm** at a constant speed of **6 m/s**.

Find volume of water filled into the container per second in litres/s, leaving your answer in terms of $\pi$.

Hence, find the total time needed for the container to be completely filled, correcting your answer to **2** decimal places.

After $t$ seconds, the depth of the water in the container is **1.4 m** and the breadth of the water surface in the container is **2.3 m**.

Find the value of $t$, correcting your answer to **2** decimal places.

Find the volume of empty space in the container at this moment.

Find the area of the water surface at this moment.

Correction to the retained source diagram: at depth 1.4 m, the width is $1.6+(2.4-1.6)(1.4/1.6)=2.3\text{ m}$, not the printed 2.1 m.

Volume and Surface Area of Prisms

Calculating Rates and Quantities

Ratio, Rate and Speed

<p>A cup of hot coffee is at <strong>90°C</strong>. The temperature of the coffee decreases at a rate of <strong>5°C</strong> per minute for the first <strong>5</strong> minutes, then at a rate of <strong>2°C</strong> per minute for the next <strong>10</strong> minutes, and finally at a rate of <strong>1°C</strong> per minute until it reaches room temperature at <strong>25°C</strong>.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Calculate the time it takes for the coffee to cool down from <strong>90°C</strong> to room temperature.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the average rate of decreasing for the coffee to cool down from <strong>90°C</strong> to <strong>25°C</strong>.</span></div></li></ol>

A cup of hot coffee is at **90°C**. The temperature of the coffee decreases at a rate of **5°C** per minute for the first **5** minutes, then at a rate of **2°C** per minute for the next **10** minutes, and finally at a rate of **1°C** per minute until it reaches room temperature at **25°C**.

Calculate the time it takes for the coffee to cool down from **90°C** to room temperature.

Calculate the average rate of decreasing for the coffee to cool down from **90°C** to **25°C**.

Rate

<p>A prism-shaped water drainage system is such that the cross section consists of a trapezium and a small rectangle as shown in the figures below.</p>
<ol class="i-list"><li><div class="list-item-paragraph">On a sunny day, only part of the rectangular part is submerged by water. Given that the water flows at a constant speed of $5$<strong>m/s</strong>, find:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/05/Question-25-i-diagram.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yNS1pLWRpYWdyYW0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY4NCwiZXhwIjoxNzkyNjYyOTg0fQ.Kx_RtCwAZOIotzK2q81YzDSBHgDuzQlXUNVomAoaOy0" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><ol class="abc-list"><li><div class="list-item-paragraph">the water-draining rate, which is given by the volume of water that flows through the cross section per unit of time in <strong>m$^3$/s</strong>, and</div></li><li><div class="list-item-paragraph">the volume of water drained in $30$ minutes, assuming that the area of the cross section of water and the speed at which water flows do not change.</div></li><li><div class="list-item-paragraph">the portion of the drainage capacity of the system used on this day, which is given by the area of the cross section of water as a percentage of the area of the cross section of the system, correcting your answer to $2$ decimal places.</div></li></ol></li><li><div class="list-item-paragraph">On a rainy day, the rectangular part is completely submerged by water and the area of the trapezoidal part of the cross section of water is <strong>$26$ m$^2$</strong>. Given that the water flows at a constant speed of <strong>$15$m/s</strong>, find:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/05/Question-25-ii-diagram.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNS9RdWVzdGlvbi0yNS1paS1kaWFncmFtLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI2NjI2ODQsImV4cCI6MTc5MjY2Mjk4NH0.DuqmzjFTUruYpIME6fBSiIufPDtqiVjRn4w2LmIVTZU" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><ol class="abc-list"><li><div class="list-item-paragraph">the water-draining rate in terms of the volume of water that flows through the cross section per unit of time in <strong>m$^3$/s</strong>, and</div></li><li><div class="list-item-paragraph">the volume of water drained in $30$ minutes, assuming that the area of the cross section of water and the speed at which water flows do not change.</div></li><li><div class="list-item-paragraph">the portion of the drainage capacity of the system used on this day, which is given by the area of the cross section of water as a percentage of the area of the cross section of the system, correcting your answer to $2$ decimal places.</div></li></ol></li></ol>

A prism-shaped water drainage system is such that the cross section consists of a trapezium and a small rectangle as shown in the figures below.

Volume and Surface Area of Composite Solids

Expressing One Quantity as a Percentage of Another

Percentages

<p>The figure below shows a container that is a triangular prism with a right-angled triangular cross section. The container is empty at first, and then is filled with water by a pipe with diameter $5$cm at a constant speed of $3$m/s.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767695624847-xzbktg02uog.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTU2MjQ4NDcteHpia3RnMDJ1b2cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY4NCwiZXhwIjoxNzkyNjYyOTg0fQ.jxawAVzKeQKYdcrQjRyUmp5_8k9R-CIp9wjc-nLj_vE" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find volume of water filled into the container per second in cm$^3$/s, leaving your answer in terms of $\pi$.</span></div></li><li><div class="list-item-paragraph">Hence, find the total time needed, in seconds, for the container to be completely filled, correcting your answer to $2$ decimal places.</div></li></ol>
<p>After $t$ seconds, the depth of the water in the container is $1$m and the length of the water surface in the container is $0.8$m.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the value of $t$, correcting your answer to $2$ decimal places.</div></li><li><div class="list-item-content"><span class="list-item-text">Find the volume of empty space in the container at this moment.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the surface area of the container that is in contact with water at this moment.</span></div></li></ol>

The figure below shows a container that is a triangular prism with a right-angled triangular cross section. The container is empty at first, and then is filled with water by a pipe with diameter $5$cm at a constant speed of $3$m/s.

Find volume of water filled into the container per second in cm$^3$/s, leaving your answer in terms of $\pi$.

After $t$ seconds, the depth of the water in the container is $1$m and the length of the water surface in the container is $0.8$m.

Find the surface area of the container that is in contact with water at this moment.

Volume and Surface Area of Cylinders

<p>Given that a water tank for irrigation has a volume of $330$ litres, with a water tap that fills water into the tank at a rate of $15$ litres per minute, and a water drain that can drain water at a rate of $30$ litres per minute during irrigation. Find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The amount of time it takes (in minutes) for the tank to be filled with water from empty to full when the water drain is blocked,</span></div></li><li><div class="list-item-content"><span class="list-item-text">The amount of time it takes (in minutes) for the tank to be drained from full to empty when the water tap is turned off,</span></div></li><li><div class="list-item-content"><span class="list-item-text">The amount of time it takes (in minutes) for the tank to be drained from full to empty when the water tap is turned on,</span></div></li><li><div class="list-item-content"><span class="list-item-text">To improve the efficiency of irrigation, a new water drain is installed, which works together with the old one. Given that it now takes $10$ minutes for the tank to be drained from full to empty if both water drains work together when the water tap is turned on, find the rate of drainage of the new water drain.</span></div></li></ol>

Given that a water tank for irrigation has a volume of $330$ litres, with a water tap that fills water into the tank at a rate of $15$ litres per minute, and a water drain that can drain water at a rate of $30$ litres per minute during irrigation. Find

The amount of time it takes (in minutes) for the tank to be filled with water from empty to full when the water drain is blocked,

The amount of time it takes (in minutes) for the tank to be drained from full to empty when the water tap is turned off,

The amount of time it takes (in minutes) for the tank to be drained from full to empty when the water tap is turned on,

To improve the efficiency of irrigation, a new water drain is installed, which works together with the old one. Given that it now takes $10$ minutes for the tank to be drained from full to empty if both water drains work together when the water tap is turned on, find the rate of drainage of the new water drain.

Combined Work and Flow Rates

<p> $5$ workers are employed for a construction project. Assume the project requires $480$ worker-hours of work, each worker works $8$ hours per day and earns an hourly rate of $\$10$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the number of days needed to complete the project and the total amount of money to be paid to the workers.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Is the total amount of money to be paid to the workers the same if the project manager employs $5$ more workers to complete the project? State your reasons with detailed calculation.</div></li><li data-list-item-id="item-1767620134618-ge9xprbwb" data-list-item-path="item-1767620134618-ge9xprbwb"><div class="list-item-paragraph">Find the least number of additional workers that to be employed in order to complete the project in $8$ days.</div></li></ol>
<p>Overtime is paid at $15$ dollars per hour for work beyond $8$ hours per day.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-1767620320212-e5bdwj0pb" data-list-item-path="item-1767620320212-e5bdwj0pb"><div class="list-item-paragraph">Find the new total amount of money to be paid to the workers if the project needs to be completed in $8$days without employing more workers, i.e., complete with $5$ workers. (Assume every worker works the same amount of time every day.)</div></li></ol>
<p>Assume every worker maintains the same constant hourly work rate, with no efficiency loss when the workforce or daily hours change.</p>

 $5$ workers are employed for a construction project. Assume the project requires $480$ worker-hours of work, each worker works $8$ hours per day and earns an hourly rate of $\$10$.

Overtime is paid at $15$ dollars per hour for work beyond $8$ hours per day.

Assume every worker maintains the same constant hourly work rate, with no efficiency loss when the workforce or daily hours change.

<p>Rajesh wants to buy some peanut butter at the grocery store, where he is offered two options.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767619754861-qyy7re8zlf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTk3NTQ4NjEtcXl5N3JlOHpsZi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjYyNjg0LCJleHAiOjE3OTI2NjI5ODR9.IB5tFU9EVoixHNMnxcswFiKDchITw4PHb8Rxcxtzf1A" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">State, with clear workings, which option has better value for money.</div></li></ol>
<p>Assume Rajesh chooses Option B and consumes a fixed amount of peanut butter every day.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">If it takes exactly $30$ days to use up all the peanut butter he buys, find the amount of peanut butter Rajesh consumes every day.</div></li><li data-list-item-id="item-1767619820806-7emomiuo0" data-list-item-path="item-1767619820806-7emomiuo0"><div class="list-item-paragraph">If Rajesh consumes $13.5$ g peanut butter every day instead, find the number of days after which all the peanut butter is used up. Find the least whole number of days by the end of which it is completely used up.</div></li></ol>

Rajesh wants to buy some peanut butter at the grocery store, where he is offered two options.

Assume Rajesh chooses Option B and consumes a fixed amount of peanut butter every day.

Comparing Unit Prices

Ratio and Rate

<p>A vehicle travels $148$ km. On average, it travels $10.3$ km for every litre of petrol.</p>
<p>Write down an expression that mentally estimates the amount of petrol used.</p>

A vehicle travels $148$ km. On average, it travels $10.3$ km for every litre of petrol.

Write down an expression that mentally estimates the amount of petrol used.

Estimation

Approximation and Estimation

<p>The capsule below is made up of a cylinder with two hemispherical bases of radius $3$ cm. The total length of the container is $16\frac{1}{3}$ cm.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Find the total volume of the capsule.</div></li><li><div class="list-item-paragraph">Find the total surface area of the capsule.</div></li><li><div class="list-item-paragraph">$135$g of liquid $A$, with density of $0.8$g/cm$^{3}$, is poured into the capsule. Find the height, $H$, of liquid $A$ in the capsule.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765256979124-88tk2g6qo7c.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNTY5NzkxMjQtODh0azJnNnFvN2MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY4NCwiZXhwIjoxNzkyNjYyOTg0fQ.iuigwWsf4bLUcn0_BqjEbZM1HlcC7sC-TXps3xEkQHc" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The capsule below is made up of a cylinder with two hemispherical bases of radius $3$ cm. The total length of the container is $16\frac{1}{3}$ cm.

Conservation of Volume and Displacement

Volume and Surface Area of a Capsule

Mensuration (Pyramid, Cones and Spheres)

<p>The graph shows the average price of petrol per litre, in the first nine months of a particular year.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1764465483930-06zgohwuc3m7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0NjU0ODM5MzAtMDZ6Z29od3VjM203LnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI2NjI2ODQsImV4cCI6MTc5MjY2Mjk4NH0.UVEm5E8aR0rEjXLh6XUuTW3EXn3z-5RK7oBsrT8o6go" style="max-width: min(450px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">State the month with the highest price of petrol.</div></li><li><div class="list-item-paragraph">Find the ratio of the price of petrol in the month of April to the month of September.</div></li><li><div class="list-item-paragraph">Express as a fraction of the first nine months, the number of months with petrol prices greater than $\$1.50$ per litre.<br></div></li><li><div class="list-item-paragraph">Find the percentage increase in the price of petrol between January and September.</div></li><li><div class="list-item-paragraph">Steven needs to go to Town $S$ in June which is $264$ km away.  Assuming his car is able to travel $12$ km on a litre of petrol, calculate the amount he will spend on petrol for the trip.</div></li><li><div class="list-item-paragraph">On a certain month, Sebastian went to a petrol kiosk and paid $\$50$ to fill his car with $40$ litres of petrol.  Find the month that Sebastian went to the petrol kiosk.</div></li></ol>

The graph shows the average price of petrol per litre, in the first nine months of a particular year.

Percentage Increase and Decrease

Simplifying Ratios

Line Graphs

Line Graph

Statistics

<p>Diagram I shows the top view of a prism in which the shaded region represents the hollow part. Diagram II shows the prism of height $16$ cm with a hollow cylindrical pipe being drilled through.<br>Given that&nbsp;$PQ=5$ cm, $QR=PS=5$ cm, $PX=4$ cm, $SR=11$ cm and $SY=16$ cm.<br>The radius of the shaded circle is $1.2$ cm.  By taking $\pi $ as&nbsp;$3.142$, calculate</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1777973507760-flzd1c665"><div class="list-item-paragraph">the perimeter of the cross-sectional area as shown in Diagram I.</div></li><li data-list-item-id="item-1777973508171-x4m9v7uui"><div class="list-item-paragraph">the area of unshaded region in Diagram I.</div></li><li data-list-item-id="item-1777973508715-wogrzd984"><div class="list-item-paragraph">the total surface area of the prism in Diagram II.</div></li><li data-list-item-id="item-1777973521824-cnl8cggt6"><div class="list-item-paragraph">the volume of the material which is used to make this prism.</div></li><li data-list-item-id="item-1777973520048-wiq3zbet1"><div class="list-item-paragraph">the cost, correct to the nearest cent, to paint the total surface area of the prism, given that the cost for painting the prism is $\$0.02$ for one square centimetre.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1764404737689-xch3iywrjd.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDQ3Mzc2ODkteGNoM2l5d3JqZC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjYyNjg0LCJleHAiOjE3OTI2NjI5ODR9.OBt7qbztL2BYGvQ8E0OUbcPXokumSS-Dl5rZAcix8m8" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Diagram I shows the top view of a prism in which the shaded region represents the hollow part. Diagram II shows the prism of height $16$ cm with a hollow cylindrical pipe being drilled through.
Given that $PQ=5$ cm, $QR=PS=5$ cm, $PX=4$ cm, $SR=11$ cm and $SY=16$ cm.
The radius of the shaded circle is $1.2$ cm.  By taking $\pi $ as $3.142$, calculate

Prism with Hollow Circular Section – Perimeter, Area & Surface Area

<p>In this question, take $\pi $ to be $3.142$.</p>
<p>The diagram below shows a solid which is made by attaching a half-cylindrical block with diameter of $8$ cm and a height of $9$ cm to a cuboid with length $8$ cm, width $5$ cm and height of $9$ cm.  The shaded region shows the cross-sectional area of the solid.</p>
<ol class="abc-list"><li data-list-item-id="item-1764402295419-7xblz5y8b" data-list-item-path="item-1764402295419-7xblz5y8b"><div class="list-item-paragraph">Find the cross-sectional area of the solid.  Give your answers to $3$ significant figures.</div></li></ol>
<div class="graph-block" data-graph-config="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" data-width="177" data-height="189" data-proportional style="width:177px;height:189px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:177px;--graph-aspect:177 / 189;"></div>
<p>A round cylindrical hole with a diameter of $2$ cm is drilled through the middle of the cuboid to form an artefact as shown.</p>
<div class="graph-block" data-graph-config="eyJheGlzIjpmYWxzZSwiZ3JpZCI6ZmFsc2UsIndpZHRoIjoxODksImhlaWdodCI6MjA2LCJlbGVtZW50cyI6W3siaWQiOiJpbXBvcnRlZC1iZXppZXItYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMCIsInR5cGUiOiJiZXppZXIiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludHMiOltbMzUwLDU4N10sWzI0OSw2MTZdLFsxNjcsNTY2XSxbMTY3LDQ5OF1dLCJmaWxsQ29sb3IiOiJub25lIiwic3Ryb2tlV2lkdGgiOjEuMn0seyJpZCI6ImltcG9ydGVkLWJlemllci1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xIiwidHlwZSI6ImJlemllciIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1sxNjcsNDk4XSxbMTY3LDQ1OF0sWzE5Myw0MjZdLFsyMzQsNDAxXV0sImZpbGxDb2xvciI6Im5vbmUiLCJzdHJva2VXaWR0aCI6MS4yfSx7ImlkIjoiaW1wb3J0ZWQtcG9seWxpbmUtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMiIsInR5cGUiOiJwb2x5bGluZSIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1syMzQsNDAxXSxbMzUwLDU4N10sWzUxMCw1MTddLFszOTEsMzI5XSxbMjM0LDQwMV1dLCJmaWxsQ29sb3IiOiJub25lIiwic3Ryb2tlV2lkdGgiOjEuMn0seyJpZCI6ImltcG9ydGVkLXNlZ21lbnQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMyIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMTY3LDQ5OF0sInBvaW50MiI6WzE2NywyNDNdLCJzdHJva2VXaWR0aCI6MS4yfSx7ImlkIjoiaW1wb3J0ZWQtYmV6aWVyLWJsb2NrLTAtY29udGVudC1lbGVtZW50LTQiLCJ0eXBlIjoiYmV6aWVyIiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnRzIjpbWzE2NywyNDNdLFsxNjcsMjAxXSxbMTkzLDE2N10sWzIzNCwxNDNdXSwiZmlsbENvbG9yIjoibm9uZSIsInN0cm9rZVdpZHRoIjoxLjJ9LHsiaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTUiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzIzNCw0MDFdLCJwb2ludDIiOlsyMzQsMTQzXSwic3Ryb2tlV2lkdGgiOjEuMn0seyJpZCI6ImltcG9ydGVkLXNlZ21lbnQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtNiIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMzkxLDMyOV0sInBvaW50MiI6WzM5MSw3MV0sInN0cm9rZVdpZHRoIjoxLjJ9LHsiaWQiOiJpbXBvcnRlZC1wb2x5bGluZS1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC03IiwidHlwZSI6InBvbHlsaW5lIiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnRzIjpbWzIzNCwxNDNdLFszOTEsNzFdLFs1MTAsMjU3XSxbNTEwLDUxN11dLCJmaWxsQ29sb3IiOiJub25lIiwic3Ryb2tlV2lkdGgiOjEuMn0seyJpZCI6ImltcG9ydGVkLWVsbGlwc2UtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtOCIsImRhc2giOjIsInR5cGUiOiJlbGxpcHNlIiwiY29sb3IiOiIjMDAwMDAwIiwiY2VudGVyIjpbMzY5LDQ1M10sInJhZGl1c1giOjQ1LCJyYWRpdXNZIjozOCwiZmlsbENvbG9yIjoibm9uZSIsInN0cm9rZVdpZHRoIjoxLjJ9LHsiaWQiOiJpbXBvcnRlZC1lbGxpcHNlLWJsb2NrLTAtY29udGVudC1lbGVtZW50LTkiLCJkYXNoIjoyLCJ0eXBlIjoiZWxsaXBzZSIsImNvbG9yIjoiIzAwMDAwMCIsImNlbnRlciI6WzM2OSwyMjddLCJyYWRpdXNYIjo0NSwicmFkaXVzWSI6NDMsImZpbGxDb2xvciI6Im5vbmUiLCJzdHJva2VXaWR0aCI6MS4yfSx7ImlkIjoiaW1wb3J0ZWQtc2VnbWVudC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xMCIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMzI1LDQ1M10sInBvaW50MiI6WzMyNSwyMjddLCJzdHJva2VXaWR0aCI6MS4yfSx7ImlkIjoiaW1wb3J0ZWQtc2VnbWVudC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xMSIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbNDE0LDQ1M10sInBvaW50MiI6WzQxNCwyMjddLCJzdHJva2VXaWR0aCI6MS4yfSx7ImlkIjoiaW1wb3J0ZWQtc2VnbWVudC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xMiIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMzM2LDQ1Ml0sInBvaW50MiI6WzQwNyw0NTJdLCJsYXN0QXJyb3ciOnRydWUsImZpcnN0QXJyb3ciOnRydWUsInN0cm9rZVdpZHRoIjoxLjJ9LHsiaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTEzIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlszODYsNDY0XSwicG9pbnQyIjpbMzg2LDYwMF0sInN0cm9rZVdpZHRoIjoxLjJ9LHsiaWQiOiJpbXBvcnRlZC1hcnJvdy1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xNCIsInR5cGUiOiJhcnJvdyIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzM4Niw2MDBdLCJwb2ludDIiOls0NzQsNjAwXSwic3Ryb2tlV2lkdGgiOjEuMn0seyJ4Ijo0ODksInkiOjU5NiwiaWQiOiJpbXBvcnRlZC10ZXh0LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTE1IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKHsyXFwsXFxtYXRocm17Y219fVxcKSJ9LHsiaWQiOiJpbXBvcnRlZC1zZWdtZW50LWJsb2NrLTAtY29udGVudC1lbGVtZW50LTE2IiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlsxNDYsNDk5XSwicG9pbnQyIjpbMTQ2LDIzN10sImxhc3RBcnJvdyI6dHJ1ZSwiZmlyc3RBcnJvdyI6dHJ1ZSwic3Ryb2tlV2lkdGgiOjEuMn0seyJ4Ijo0NiwieSI6MzgxLjEsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xNyIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCh7OVxcLFxcbWF0aHJte2NtfX1cXCkifSx7ImlkIjoiaW1wb3J0ZWQtc2VnbWVudC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0xOCIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbMjM3LDExNV0sInBvaW50MiI6WzM5Miw0N10sImxhc3RBcnJvdyI6dHJ1ZSwiZmlyc3RBcnJvdyI6dHJ1ZSwic3Ryb2tlV2lkdGgiOjEuMn0seyJ4IjoyNTQsInkiOjYxLCJpZCI6ImltcG9ydGVkLXRleHQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMTkiLCJ0eXBlIjoidGV4dCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoezVcXCxcXG1hdGhybXtjbX19XFwpIn0seyJpZCI6ImltcG9ydGVkLXNlZ21lbnQtYmxvY2stMC1jb250ZW50LWVsZW1lbnQtMjAiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6WzQxMSw1NF0sInBvaW50MiI6WzUyNywyMzVdLCJsYXN0QXJyb3ciOnRydWUsImZpcnN0QXJyb3ciOnRydWUsInN0cm9rZVdpZHRoIjoxLjJ9LHsieCI6NDgwLCJ5IjoxMzAsImlkIjoiaW1wb3J0ZWQtdGV4dC1ibG9jay0wLWNvbnRlbnQtZWxlbWVudC0yMSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCh7OFxcLFxcbWF0aHJte2NtfX1cXCkifV0sImFsaWdubWVudCI6ImNlbnRlciIsInhBeGlzTGFiZWwiOiJcXCh4XFwpIiwieUF4aXNMYWJlbCI6IlxcKHlcXCkiLCJib3VuZGluZ0JveCI6WzU2LjE1NTUsNjA0LjI3ODEsNTQ2LjUzOSw2Ni4wNDE3XSwib3JpZ2luTGFiZWwiOiJcXChPXFwpIiwiZ3JpZFNldHRpbmdzIjp7Im1ham9yQ29sb3IiOiIjOTRhM2I4IiwibWFqb3JTdGVwWCI6MSwibWFqb3JTdGVwWSI6MSwibWlub3JDb2xvciI6IiNjYmQ1ZTEiLCJtYWpvck9wYWNpdHkiOjAuNDUsIm1pbm9yT3BhY2l0eSI6MC40NSwibWlub3JFbGVtZW50c1giOjQsIm1pbm9yRWxlbWVudHNZIjo0LCJtYWpvclN0cm9rZVdpZHRoIjowLjgsIm1pbm9yU3Ryb2tlV2lkdGgiOjAuNSwic2hvd01ham9yR3JpZExpbmVzIjp0cnVlLCJzaG93TWlub3JHcmlkTGluZXMiOnRydWV9LCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlLCJ4VGlja0xhYmVsRXZlcnkiOjEsInlUaWNrTGFiZWxFdmVyeSI6MSwieEF4aXNMYWJlbE9mZnNldCI6WzIsLTExXSwieUF4aXNMYWJlbE9mZnNldCI6Wy0xNCw1XSwib3JpZ2luTGFiZWxPZmZzZXQiOlstNSwtMl19" data-width="189" data-height="206" data-proportional style="width:189px;height:206px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:189px;--graph-aspect:189 / 206;"></div>
<ol class="abc-list" start="2"><li data-list-item-id="item-1789461101736-1wn4xam0x" data-list-item-path="item-1789461101736-1wn4xam0x"><div class="list-item-paragraph"><p>Find the volume of the artefact. Give your answer correct to $3$ significant figures.</p></div></li></ol>
<p>The artefact needs to be applied with a protective layer.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1764402374101-m3ywo30i0" data-list-item-path="item-1764402374101-m3ywo30i0"><div class="list-item-paragraph">Given that the protective layer costs $\$0.50$ per cm$^{2}$, find the total cost of applying the layer.</div></li></ol>

In this question, take $\pi $ to be $3.142$.

The diagram below shows a solid which is made by attaching a half-cylindrical block with diameter of $8$ cm and a height of $9$ cm to a cuboid with length $8$ cm, width $5$ cm and height of $9$ cm.  The shaded region shows the cross-sectional area of the solid.

A round cylindrical hole with a diameter of $2$ cm is drilled through the middle of the cuboid to form an artefact as shown.

The artefact needs to be applied with a protective layer.

Composite Solid with Half-Cylinder and Cylindrical Hole

<p>The diagram shows the cross section of a cylindrical pipe with radius $5.6$ m. The shaded region shows the thickness of the pipe, which is $2.8$ m.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1777886879997-x0a6jhwn2" data-list-item-path="item-1777886879997-x0a6jhwn2"><div class="list-item-paragraph">Find the area of the shaded region.</div></li><li data-list-item-id="item-1777886885629-yl3gh0qk5" data-list-item-path="item-1777886885629-yl3gh0qk5"><div class="list-item-paragraph">The density of the material used to make the pipe is $40$ kg/m${^3}$, find the mass of the $30$ m pipe.  Give your answer correct to $3$ significant figures.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1764400386836-vm9kz48v21f.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDAzODY4MzYtdm05a3o0OHYyMWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY4NCwiZXhwIjoxNzkyNjYyOTg0fQ.-QQeA8_7keo-BOvSrdaME_3-9hnKyj2Vi5aiR5WebJM" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows the cross section of a cylindrical pipe with radius $5.6$ m. The shaded region shows the thickness of the pipe, which is $2.8$ m.

Hollow Cylinder (Pipe) – Area and Mass Calculation

<p>A piece of molten iron is at $1200{}^\circ \mathrm{C}$. The temperature of the metal falls at a rate of $65{}^\circ \mathrm{C}$ per second for the first $10$ seconds, at a rate of $30{}^\circ \mathrm{C}$ per second for the next $8$ seconds and then at a rate of $3{}^\circ \mathrm{C}$ per second until it reaches room temperature of $27{}^\circ \mathrm{C}$. Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the temperature of the metal after $12$ seconds,</div></li><li><div class="list-item-paragraph">the total time needed for to reach a temperature of $27{}^\circ \mathrm{C}$ from $1200{}^\circ \mathrm{C}$.</div></li></ol>

A piece of molten iron is at $1200{}^\circ \mathrm{C}$. The temperature of the metal falls at a rate of $65{}^\circ \mathrm{C}$ per second for the first $10$ seconds, at a rate of $30{}^\circ \mathrm{C}$ per second for the next $8$ seconds and then at a rate of $3{}^\circ \mathrm{C}$ per second until it reaches room temperature of $27{}^\circ \mathrm{C}$. Find

Rate of Change Problem: Cooling in Stages

<p>Find the cost of fencing a rectangular plot of land of dimension $5.2$ m long and $6.3$ m wide using wire of unit price $\$4.50$ /m.</p>

Find the cost of fencing a rectangular plot of land of dimension $5.2$ m long and $6.3$ m wide using wire of unit price $\$4.50$ /m.

Perimeter and Area of Rectangles and Squares

Cost of Fencing a Rectangular Plot

Perimeter and Area of Plane Figures

<p>A printer can print $125$ sheets of paper in $2$ minutes.  How many sheets of paper can the printer print in $1$ hour?</p>

A printer can print $125$ sheets of paper in $2$ minutes.  How many sheets of paper can the printer print in $1$ hour?

Rate Word Problem: Printing per Hour

<p>The land area of Singapore is $712.4$ km². The population of Singapore in the year $2010$ was $5076700$. On average, how many people lived on $1$ km² of land? Give your answer correct to $3$ significant figures.</p>

The land area of Singapore is $712.4$ km². The population of Singapore in the year $2010$ was $5076700$. On average, how many people lived on $1$ km² of land? Give your answer correct to $3$ significant figures.

Application of Significant Figures

<p>The figure below shows a swimming pool that consists of two parts, the shallow cuboid part, where the depth is $1.2$ m, and the deep trapezoidal-prism part, where the depth is from $1.2$ m to $3$ m. The two parts are equal in length ($30$ m) and breadth ($9$ m), and the two parts are connected such that water is filled and drained simultaneously into and out of the pool.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1777978100597-38b6630yjdm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzc5NzgxMDA1OTctMzhiNjYzMHlqZG0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY4NCwiZXhwIjoxNzkyNjYyOTg0fQ.LNdQ_lN74XbdSy5LO7EtuDqcMcO2m7T2Ix6vFRFjlx8" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">Find the volume of the shallow cuboid part and the deep trapezoidal-prism part, respectively. Hence, find the total volume of the whole swimming pool.</div></li><li><div class="list-item-paragraph">Given that the whole swimming pool is completely filled with water originally, for maintenance need, water is drained out at a constant rate of $9$ m$^{3}$ per minute. Find</div><ol class="abc-list"><li><div class="list-item-paragraph">the amount of time after which the shallow cuboid part is completely drained,</div></li><li><div class="list-item-paragraph">the amount of time after which the whole swimming pool is completely drained, and</div></li><li><div class="list-item-paragraph">the depth of water (measured from the bottom of the deep part) after $36$ minutes.</div></li><li><div class="list-item-paragraph">Determine the type of the shape formed by the remaining water after $90$ minutes.</div></li></ol></li><li><div class="list-item-paragraph">After maintenance, water is refilled into the empty swimming pool at a constant rate of $9$ m$^{3}$ per minute. In the grid below, draw the graph of the depth, $h$, of water with respect to time, $t$. Approximately sketch the curved part and accurately draw the straight-line part of the graph, if there is any.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1777978440480-7w0t1whzcg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzc5Nzg0NDA0ODAtN3cwdDF3aHpjZy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjYyNjg0LCJleHAiOjE3OTI2NjI5ODR9.KLlMCmKPLi0BY9alvZizGF8oniXx2kA2BOP92d6MJsM" alt="List item image" class="aligncenter custom-width" style="max-width: min(600px, 100%); height: auto;" /></div></li></ol>

The figure below shows a swimming pool that consists of two parts, the shallow cuboid part, where the depth is $1.2$ m, and the deep trapezoidal-prism part, where the depth is from $1.2$ m to $3$ m. The two parts are equal in length ($30$ m) and breadth ($9$ m), and the two parts are connected such that water is filled and drained simultaneously into and out of the pool.

Water-Level Graphs at Constant Volume Flow

Linear Graphs in Real Life Problems

IP1U13 HOT Q9

Functions and Graphs

Linear Graphs and Its Applications

<p>The figure shows a container that is a trapezoidal prism. The container is empty at first, and then is filled with water by a pipe with diameter $10$ cm at a constant speed of $6$$\mathrm{m/s}$.</p>
<ol class="abc-list"><li data-list-item-id="item-1780822247027-i43ltc4gl" data-list-item-path="item-1780822247027-i43ltc4gl"><div class="list-item-paragraph">Find volume of water filled into the container per second in litres/s, leaving your answer in terms of $\pi $.</div></li><li data-list-item-id="item-1780822247611-knl4tekzi" data-list-item-path="item-1780822247611-knl4tekzi"><div class="list-item-paragraph">Hence, find the total time needed for the container to be completely filled, correcting your answer to $2$ decimal places.</div></li></ol>
<p>After $t$ seconds, the depth of the water in the container is $1.4$ m and the breadth of the water surface is $2.3$ m.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="3"><li data-list-item-id="item-1777977559509-s3dn7h3ew" data-list-item-path="item-1777977559509-s3dn7h3ew"><div class="list-item-paragraph">Find the value of $t$, correcting your answer to $2$ decimal places.</div></li><li data-list-item-id="item-1777977560073-l4s6p7sxg" data-list-item-path="item-1777977560073-l4s6p7sxg"><div class="list-item-paragraph">Find the volume of empty space in the container at this moment.</div></li><li data-list-item-id="item-1777977560953-1hi7yq156" data-list-item-path="item-1777977560953-1hi7yq156"><div class="list-item-paragraph">Find the area of the water surface at this moment.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1777977742938-z5w5etctuff.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzc5Nzc3NDI5MzgtejV3NWV0Y3R1ZmYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY2MjY4NCwiZXhwIjoxNzkyNjYyOTg0fQ.m4k0jYTRuPT9Y8MM1NzTaAs0rb_DwPLctdyOFVideSw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>
<p>Source correction: the diagram’s $2.1\text{ m}$ label is inconsistent with its other dimensions. The width at height $1.4\text{ m}$ is $1.6+(2.4-1.6)(1.4/1.6)=2.3\text{ m}$. Use this corrected width in parts (c)–(e).</p>

The figure shows a container that is a trapezoidal prism. The container is empty at first, and then is filled with water by a pipe with diameter $10$ cm at a constant speed of $6$$\mathrm{m/s}$.

After $t$ seconds, the depth of the water in the container is $1.4$ m and the breadth of the water surface is $2.3$ m.

Source correction: the diagram’s $2.1\text{ m}$ label is inconsistent with its other dimensions. The width at height $1.4\text{ m}$ is $1.6+(2.4-1.6)(1.4/1.6)=2.3\text{ m}$. Use this corrected width in parts (c)–(e).

IP1U13 HOT Q8

<p>Water runs through a pipe which has a diameter of $42$ cm at a constant rate of $5$ m/s.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the volume of water that runs through the pipe in every second is $693\mathrm{ }000$ cm$^{3}$, correct to $3$ significant figures.</div></li><li><div class="list-item-paragraph">If a rectangular diving pool measuring $20$ m by $4$ m by $8$ m is to be filled with water, find the time taken, in minutes, to fill two-thirds of the pool.</div></li></ol>

Water runs through a pipe which has a diameter of $42$ cm at a constant rate of $5$ m/s.

Math Question Bank