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PSLE

PSLE Foundation Mathematics

<p>The figure below shows a tilted cuboid water tank with dimensions <strong>25 cm</strong> by <strong>20 cm</strong> by <strong>80 cm</strong>, such that the water in the tank forms a triangular prism with base triangle $BCM$. Given that $M$ and $N$ are midpoints of sides $CD$ and $EF$ of the cuboid tank, respectively.</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/1767771757170-8qtgpx3ubge.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzE3NTcxNzAtOHF0Z3B4M3ViZ2UucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.9ntIyYbaTR2KRHBbgk7u97L10MF-1vu6C97OCEEMj3s" style="max-width: min(300px, 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 the volume of the water in the tank.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the surface area of the tank that is in contact with water.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the depth of water when the tank is placed in such a way that face $ABCD$ is the base.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the depth of water when the tank is placed in such a way that face $CDFE$ is the base.</span></div></li></ol>

The figure below shows a tilted cuboid water tank with dimensions **25 cm** by **20 cm** by **80 cm**, such that the water in the tank forms a triangular prism with base triangle $BCM$. Given that $M$ and $N$ are midpoints of sides $CD$ and $EF$ of the cuboid tank, respectively.

Find the volume of the water in the tank.

Find the surface area of the tank that is in contact with water.

Find the depth of water when the tank is placed in such a way that face $ABCD$ is the base.

Find the depth of water when the tank is placed in such a way that face $CDFE$ is the base.

Volume and Surface Area of Prisms

Conservation of Volume and Displacement

<p>An open cylindrical can with diameter <strong>6 cm</strong> and height <strong>10 cm</strong> is used to fill water into an empty cuboid tank with dimensions <strong>48 cm</strong> by <strong>24 cm</strong> by $h$ <strong>cm</strong>. Assume the thickness of all containers is negligible.</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/1767771162674-ejcfdib5s1o.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzExNjI2NzQtZWpjZmRpYjVzMW8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.wzwRgygxhwogicCrEST_XKcMMzlER3TJzdtd0nE-kw8" style="max-width: min(400px, 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 the volume of the cylindrical can, 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">Given that exactly <strong>128</strong> cans of water are needed to completely fill the empty tank, find the value of $h$, leaving your answer in terms of $\pi$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$h =$ <strong>40 cm</strong>, find the minimum number of cans of water needed to completely fill the empty tank, rounding up to the next whole number if necessary.</div></li></ol>

An open cylindrical can with diameter **6 cm** and height **10 cm** is used to fill water into an empty cuboid tank with dimensions **48 cm** by **24 cm** by $h$ **cm**. Assume the thickness of all containers is negligible.

Find the volume of the cylindrical can, leaving your answer in terms of $\pi$.

Given that exactly **128** cans of water are needed to completely fill the empty tank, find the value of $h$, leaving your answer in terms of $\pi$.

Given that $h =$ **40 cm**, find the minimum number of cans of water needed to completely fill the empty tank, rounding up to the next whole number if necessary.

Volume and Surface Area of Cylinders

Volume and Surface Area of Cylinder

<p>A cuboid water tank with dimensions $12$ cm by $12$ cm by $32$ cm contains water to $\frac{3}{4}$ its height and is used to fill water to a number of identical open cylindrical cans with diameter $4$ cm and height $h$ cm. Assume the thickness of all containers is negligible.</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/1767694925729-e3sgrgfwk9f.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ5MjU3MjktZTNzZ3JnZndrOWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.gCLY8tDIDvibk_Gqr8E1WLqFmHgJ0B1bTYg8UPOTDtg" style="max-width: min(300px, 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 the volume of water contained in the cuboid tank.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Given that the maximum number of cylinder cans that can be completely filled is $8$, find the integer value of $h$. Hence, find the volume of remaining water in the tank.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$h = 4$ cm, find the maximum number of cylinder cans that can be completely filled. Hence, find the volume of remaining water in the tank.</div></li></ol>
<p>Part (b) does not specify a unique integer height. The full condition allows $h=31,32,33$ or $34\text{ cm}$; the solution lists the corresponding remainders. A request for the greatest possible integer height would select $34\text{ cm}$.</p>

A cuboid water tank with dimensions $12$ cm by $12$ cm by $32$ cm contains water to $\frac{3}{4}$ its height and is used to fill water to a number of identical open cylindrical cans with diameter $4$ cm and height $h$ cm. Assume the thickness of all containers is negligible.

Find the volume of water contained in the cuboid tank.

Given that $h = 4$ cm, find the maximum number of cylinder cans that can be completely filled. Hence, find the volume of remaining water in the tank.

Part (b) does not specify a unique integer height. The full condition allows $h=31,32,33$ or $34\text{ cm}$; the solution lists the corresponding remainders. A request for the greatest possible integer height would select $34\text{ cm}$.

Problem Solving Involving Inequalities

Linear Equations and Inequalities

Mensuration (Pyramid, Cones and Spheres)

<p>In a restaurant, the fruit punch is mixed in a container with no lid as shown in Figure 1. The container is a prism whose cross-section is a trapezium. The lengths of the parallel sides of the trapezium are $10$ cm and $15$ cm and the distance between the parallel sides is $9$ cm. It is given that the capacity of the container is $1.8$ litres.</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/1765261978122-37qq3cy9096.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjE5NzgxMjItMzdxcTNjeTkwOTYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.XAShVuvh3rrbTmeM76PTTHU-hezmFWynzwbFCtIe4b4" style="max-width: min(370px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the height of the container is $16$ cm.</div></li></ol>
<p>All the fruit punch in one full container is transferred to a jar as shown in Figure 2.</p>
<p>The jar is made by joining a hemisphere of radius $8$ cm to a part of a right circular cone of height, $h$ cm.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Given that the slant height of the cone is $17$ cm, show that&nbsp;$h=15$ cm.</div></li><li><div class="list-item-paragraph">Find the depth of the fruit punch in this jar after all the fruit punch has been transferred 	from the container in Figure 1.</div></li></ol>

In a restaurant, the fruit punch is mixed in a container with no lid as shown in Figure 1. The container is a prism whose cross-section is a trapezium. The lengths of the parallel sides of the trapezium are $10$ cm and $15$ cm and the distance between the parallel sides is $9$ cm. It is given that the capacity of the container is $1.8$ litres.

All the fruit punch in one full container is transferred to a jar as shown in Figure 2.

The jar is made by joining a hemisphere of radius $8$ cm to a part of a right circular cone of height, $h$ cm.

Volume and Surface Area of Composite Solids

Volume Transfer Between Composite Solids

<p>The diagram below shows a solid sphere. The radius of the sphere is $14$ cm. A cone of radius <br>$10$ cm and height $11.5$ cm was cut out from the sphere.<br></p>
<ol class="abc-list"><li><div class="list-item-paragraph">What is the total surface area of the cone that was cut out from the sphere?</div></li><li><div class="list-item-paragraph">What is the volume of the cone?</div></li><li><div class="list-item-paragraph">The remaining part of the sphere was moulded into a hemisphere.</div><ol class="i-list"><li><div class="list-item-paragraph">What is the radius of the hemisphere?</div></li><li><div class="list-item-paragraph">Find the total surface area of the hemisphere.</div></li></ol></li></ol>
<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/1765261372968-3d30pibpx2w.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjEzNzI5NjgtM2QzMHBpYnB4MncucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.cXkAIWSi09aruyrWgri78fiX1AC9h7YN8pXF_24n2b4" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

The diagram below shows a solid sphere. The radius of the sphere is $14$ cm. A cone of radius 
$10$ cm and height $11.5$ cm was cut out from the sphere.


Volume and Surface Area of Cones

Composite Solid: Sphere → Cone Removal → Hemisphere

<p>A metal structure formed by a cone of radius $15$ cm and height $34$ cm and a cylindrical base of the same radius and height $h$ cm is shown below.</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">Given that the volume of the structure is $15080$ cm$^{3}$, find the height, $h$ cm of the cylindrical base.</div></li><li><div class="list-item-paragraph">The solid is melted and recast into $30$ identical hemispherical shapes. Find the radius of each hemisphere.</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/1765260959599-okf6hwhyp1h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjA5NTk1OTktb2tmNmh3aHlwMWgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.fzhJ2MTY3pqeWztKqJxn2bXEg4wxSFDbGuv6YMhLKf8" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A metal structure formed by a cone of radius $15$ cm and height $34$ cm and a cylindrical base of the same radius and height $h$ cm is shown below.

Volume and Melting Application of a Composite Solid

<p>A hollow hemispheric container has an external radius of $15$ cm and an internal radius $12$ 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;"><p>Given that&nbsp;$\pi =3.142$,</p><ol class="abc-list"><li><div class="list-item-paragraph">find the total surface area of the container.</div></li><li><div class="list-item-paragraph">find the volume of the container.</div></li><li><div class="list-item-paragraph">If the hemispheric container is melted and recast into small cubes of length $2$ cm on each side. Find the maximum number of cubes which can be made.</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/1765260560487-k69ebacwypa.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjA1NjA0ODctazY5ZWJhY3d5cGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.L9UgvsrxfO248TRI5rCOxQhkRfXF6oK9zbWahBjV7g8" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A hollow hemispheric container has an external radius of $15$ cm and an internal radius $12$ cm.

Hollow Hemisphere: Surface Area and Volume

<p>The iron solid is made up of a hemisphere joined to a cylinder with a radius of $6$ cm and a height of $10$ cm as shown.</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">Calculate</div><ol class="i-list"><li><div class="list-item-paragraph">the total volume of the solid, </div></li><li><div class="list-item-paragraph">the total surface area of the solid.</div></li></ol></li><li><div class="list-item-paragraph">If the solid is melted to form a pyramid of a base area of $250$ cm$^{2}$, find the height of the pyramid.</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/1765260054310-ofpdiqfhk2i.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjAwNTQzMTAtb2ZwZGlxZmhrMmkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.JweF2Z_eTnUAOdsdsd9gdF3qM5EVYgB6MMcTNBk04a4" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The iron solid is made up of a hemisphere joined to a cylinder with a radius of $6$ cm and a height of $10$ cm as shown.

Volume and Surface Area of Pyramids

Composite Solid: Hemisphere Joined to a Cylinder

<p>A solid gold pyramid as shown below is melted to form a sphere. Find the radius of the sphere.</p>
<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/1765257690813-4pxs7i20a1e.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNTc2OTA4MTMtNHB4czdpMjBhMWUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.oGMZWHJEt7jDn474v3T4hN943tC84TfwQXFwpLLcW6I" style="max-width: min(230px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

A solid gold pyramid as shown below is melted to form a sphere. Find the radius of the sphere.

Volume Conversion: Pyramid Melted into a Sphere

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNTY5NzkxMjQtODh0azJnNnFvN2MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.-gT_hJEf8j_NIbLpvYcjW6fLBiiUm7QfRXw2FcF_MqY" 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.

Calculating Rates and Quantities

Volume and Surface Area of a Capsule

Ratio, Rate and Speed

<p>$35$ solid hemispheres, each of diameter $1.8$ cm, are melted to form a single sphere. Find the radius of the sphere.</p>

$35$ solid hemispheres, each of diameter $1.8$ cm, are melted to form a single sphere. Find the radius of the sphere.

Volume Conservation: Hemispheres Melted into a Sphere

<p>$60$ solid spheres, each of diameter $3$ cm, are melted to form a single hemisphere. Find the radius of the hemisphere.</p>

$60$ solid spheres, each of diameter $3$ cm, are melted to form a single hemisphere. Find the radius of the hemisphere.

Solid Spheres Melted to Form a Hemisphere

<p>A bowl with a hemispherical shape has a diameter of $7.8$ cm is fully filled with liquid. The liquid is then poured into a cylindrical container of diameter $5.3$ cm. Find the depth of the liquid in the container. </p>

A bowl with a hemispherical shape has a diameter of $7.8$ cm is fully filled with liquid. The liquid is then poured into a cylindrical container of diameter $5.3$ cm. Find the depth of the liquid in the container. 

Hemispherical Bowl to Cylinder: Volume Conversion

<p>The solid consists of a trapezoidal prism with a cylindrical hole of radius $10$ cm in the centre. Given that&nbsp;$AB=25$ cm, $EF=75$ cm, $FG=28$ cm, $CF=DE=93.41$ cm and the height of the solid is $90$ 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">Calculate the total volume of the solid.</div></li><li><div class="list-item-paragraph">Calculate the total surface area of the solid.</div></li><li><div class="list-item-paragraph">The solid is melted and recast into a cylindrical solid of height $40$ cm, find the radius of the solid.</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/1764407131591-smiggghjbg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDcxMzE1OTEtc21pZ2dnaGpiZy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODUwOTQzLCJleHAiOjE3OTI4NTEyNDN9.Kpch6DoxtR_rvCmUMRrleKaaKvEGsGrPDhvKG9lbgJk" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The solid consists of a trapezoidal prism with a cylindrical hole of radius $10$ cm in the centre. Given that $AB=25$ cm, $EF=75$ cm, $FG=28$ cm, $CF=DE=93.41$ cm and the height of the solid is $90$ cm.

Trapezoidal Prism with Cylindrical Hole – Volume & Surface Area

<p>A solid consists of a cuboid, to which a quadrant of a cylinder is attached as shown in the diagram below. $AE=20$ cm, $ED=30$ cm, $DC=20$ cm and $CR=70$ 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;"><p>Calculate</p><ol class="abc-list"><li><div class="list-item-paragraph">the volume of the solid,</div></li><li><div class="list-item-paragraph">the total surface area of the solid.</div></li></ol><p>The solid prism is then moulded into a solid cylinder of radius $30$cm.</p><ol class="abc-list" start="3"><li><div class="list-item-paragraph">Calculate the height of the cylinder.</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/1764406778899-7p1rvsvdlpo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDY3Nzg4OTktN3AxcnZzdmRscG8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.33CXYfJH8DiwtU_HsSWRQoyZYJKPytYnrcnagaQF358" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A solid consists of a cuboid, to which a quadrant of a cylinder is attached as shown in the diagram below. $AE=20$ cm, $ED=30$ cm, $DC=20$ cm and $CR=70$ cm.

Composite Solid: Cuboid with Attached Cylindrical Quadrant

<ol class="abc-list"><li data-list-item-id="item-1764406418197-32xx4rl8e" data-list-item-path="item-1764406418197-32xx4rl8e"><div class="list-item-paragraph">In the diagram, tank $X$ is filled with water to a height of $16$ cm. Find the volume of water in tank $X$.<br></div></li><li data-list-item-id="item-1764406418835-9r4w30j8n" data-list-item-path="item-1764406418835-9r4w30j8n"><div class="list-item-paragraph">Water in tank $X$ is poured into $2$ other rectangular tanks, $Y$ and $Z$, such that the heights of the water in the $3$ tanks are the same. Find the new height of the water in tank $X$.</div></li></ol>
<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/1764406475046-qv3ijlhxicg.JPG?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDY0NzUwNDYtcXYzaWpsaHhpY2cuSlBHIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.ELOBS035MVP43Ma7RDj8aKaDpdweglC8ioBM1n2FT0g" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Volume and Water Transfer Between Rectangular Tanks

<p>Figure $1$ shows an open trough in the form of a trapezoidal prism. $AD=HG=100$ cm, $BC=EF=50$ cm, $JE=80$ cm, $AB=DC=GF=HE=83.82$ cm and $CF=200$ cm.<br>The trough is filled to the brim with water.</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-1777973855678-ycoehr6zg" data-list-item-path="item-1777973855678-ycoehr6zg"><div class="list-item-paragraph">Calculate the volume of the water in the trough.</div></li><li data-list-item-id="item-1777973856225-nfa6avx6z" data-list-item-path="item-1777973856225-nfa6avx6z"><div class="list-item-paragraph">The trough is to be painted on its external surfaces, calculate the total area of the external surfaces to be painted.</div></li><li data-list-item-id="item-1777973856729-cqp4iufwt" data-list-item-path="item-1777973856729-cqp4iufwt"><div class="list-item-paragraph">All the water from the trough is then transferred into an open cylindrical container as shown in Figure $2$. The base of this cylinder has a circumference of $440$ cm. <br>Calculate the height of the water, $h$, in the cylinder.  $\left( \mathrm{Take }\pi =\frac{22}{7} \right)$.</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/1764406038175-i5leun4aq7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDYwMzgxNzUtaTVsZXVuNGFxNy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODUwOTQzLCJleHAiOjE3OTI4NTEyNDN9.a9PDxi7mgeCMFAtmi2RsCyGpxgF0rHvMKFZGe6i4dXQ" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Figure $1$ shows an open trough in the form of a trapezoidal prism. $AD=HG=100$ cm, $BC=EF=50$ cm, $JE=80$ cm, $AB=DC=GF=HE=83.82$ cm and $CF=200$ cm.
The trough is filled to the brim with water.

Volume and Surface Area of a Trapezoidal Trough

<p>The following water tank is filled to the brim with water.</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-1780821919439-ezodvq4oq"><div class="list-item-paragraph">Find the volume of water in the tank.</div></li><li data-list-item-id="item-1780821920711-co1lzbx87"><div class="list-item-paragraph">All the water in the tank is poured into cylindrical cups of radius $3$ cm and height $3$ cm.  Find the volume of one cup.</div></li><li data-list-item-id="item-1780821921655-2de5ncueg"><div class="list-item-paragraph">Hence, find the minimum number of cups required to contain all the water in the tank.</div></li><li data-list-item-id="item-1780821922119-bb94trdix"><div class="list-item-paragraph">If the water in the tank is to fill up exactly $4$ identical containers of uniform cross-	sectional area of $31.5$ cm$^{2}$, find the height of one such container.</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/1764404124078-wpah0dytqfa.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDQxMjQwNzgtd3BhaDBkeXRxZmEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.lNhbIuRalvu1dzxez-NMezK8qLd61RpurppnqKBYXsc" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The following water tank is filled to the brim with water.

Volume of a Water Tank and Related Applications

<p>The diagram shows the uniform cross-section of a solid paper weight $ABCDE$, which is in the shape of a trapezium with a semi-circular hole $ABC$ cut out of it. It is given that&nbsp;$AC$ is parallel to $DE$, $AC=7$ cm, $CD=15$ cm, $DE=21$ cm, $AE=13$ cm, and the height of the trapezium is $12$ 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 data-list-item-id="item-1780821806157-qz2twa1jg"><div class="list-item-paragraph">Find the area of $ABCDE$.</div></li><li data-list-item-id="item-1780821806669-1af3rdsqx"><div class="list-item-paragraph">The length of the prism is $30$ cm. Calculate its</div><ol class="i-list"><li data-list-item-id="subitem-1780821814846-gtrgu3cj8"><div class="list-item-paragraph">volume.</div></li><li data-list-item-id="subitem-1780821814965-9ht57lfyc"><div class="list-item-paragraph">surface area.</div></li></ol></li><li data-list-item-id="item-1780821807501-4lutpdjz3"><div class="list-item-paragraph">The prism is melted down and recast into a cylinder of radius $8$ cm. Find the height of 	the cylinder.</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/1764403676855-2wo3c92nwrw.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDM2NzY4NTUtMndvM2M5Mm53cncucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ._SICJzT2_y8sFA-X71uOz5tyUcXicU8Ep9OYc8pyOfI" style="max-width: min(350px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows the uniform cross-section of a solid paper weight $ABCDE$, which is in the shape of a trapezium with a semi-circular hole $ABC$ cut out of it. It is given that $AC$ is parallel to $DE$, $AC=7$ cm, $CD=15$ cm, $DE=21$ cm, $AE=13$ cm, and the height of the trapezium is $12$ cm.

Composite Cross-Section: Trapezium with Semi-Circular Cut-Out

<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;"><p>A cylindrical container with base diameter $12$ cm and height $15$ cm was initially full of water. Find the volume of remaining water in the container if the container is tilted as shown in the diagram. Leave your answer in terms of $\pi $.</p></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/1777977050876-rr386n1poyd.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzc5NzcwNTA4NzYtcnIzODZuMXBveWQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg1MDk0MywiZXhwIjoxNzkyODUxMjQzfQ.zf_88zPjJwZ0VwOSVlo-Lke5RfH0clykSkrvWvXOEFQ" style="max-width: min(280px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

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