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PSLE

PSLE Foundation Mathematics

<p>A cylindrical solid has a hemispherical top of the same radius $r\text{ cm}$. The cylinder has height $2r\text{ cm}$, and the volume of the entire solid is $576\pi\text{ cm}^3$.</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/1789189430874-y1vjytfj43.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODkxODk0MzA4NzQteTF2anl0Zmo0My5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTMwMTE2LCJleHAiOjE3OTM1MzA0MTZ9.2atMc4E2GNj4aOHyUxtKQwaUcppGgYE1qr0wE3Sek6g" style="max-width: min(150px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i1" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i1"><div class="list-item-paragraph">Show that&nbsp;$r=6$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="sec4-em-2025-mgs-s-p2-q7-b3-i2" data-list-item-path="sec4-em-2025-mgs-s-p2-q7-b3-i2"><div class="list-item-paragraph">The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.<span class="inline-marks">[4]</span></div></li></ol>

A cylindrical solid has a hemispherical top of the same radius $r\text{ cm}$. The cylinder has height $2r\text{ cm}$, and the volume of the entire solid is $576\pi\text{ cm}^3$.

The entire solid is melted to form a solid cone whose radius and height are equal. Find the total surface area of the cone.

Volume and Surface Area of Cylinders

Applications of Pythagoras’ Theorem

Volume and Surface Area of Spheres

Volume and Surface Area of Cones

Volume and Surface Area of Composite Solids

2025 MGS(S) PRELIMS P2 Q7

Volume and Surface Area of Prisms

Mensuration (Pyramid, Cones and Spheres)

Pythagoras’ Theorem

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzExNjI2NzQtZWpjZmRpYjVzMW8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.A7EFkZC29MwxHQdKuPRMR-LdWIWepCn_4tHerJ9SnlY" 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.

Conservation of Volume and Displacement

Volume and Surface Area of Cylinder

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTU2MjQ4NDcteHpia3RnMDJ1b2cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.P5GmKczJQiccvnhzcGgngMc8DRATR5LqqGyeGhh-z74" 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 volume of empty space in the container at this moment.

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

Calculating Rates and Quantities

Ratio, Rate and Speed

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ5MjU3MjktZTNzZ3JnZndrOWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.Y8_bb0RGCjz1DYR_HML6YFadfBjP2bVM-ElXDyhJpKQ" 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

<p>The figure below shows a cylinder that has a volume of $208\pi$ cm$^3$.</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/1767693559374-spfhsw9dnb9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTM1NTkzNzQtc3BmaHN3OWRuYjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.Fcem6RGYfhjmFiAzzG3B9TqIC4fZMIEAvwiXhPeu41g" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $r$, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">its total surface area.</span></div></li></ol>

The figure below shows a cylinder that has a volume of $208\pi$ cm$^3$.

<p>The figure below shows a cylinder. </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/1767693427605-dbyx5stiwfg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTM0Mjc2MDUtZGJ5eDVzdGl3ZmcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.ILSjOxl2p6k-OnB0M3sAhr0j9hZGV-M68HIeC7i04DA" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">its total surface area, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">its volume.</span></div></li></ol>

<p>An open container is made by joining a cylinder of radius $5$ cm and a cone of height $12$ cm. The container is completely filled with water. If the height of the cylinder is $9$ cm, give your answers in terms of $\pi $, where applicable.<br>Find the</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-1778406102701-mey3z7utv" data-list-item-path="item-1778406102701-mey3z7utv"><div class="list-item-paragraph">volume of the water,</div></li><li data-list-item-id="item-1778406103369-6t1dwhbsk" data-list-item-path="item-1778406103369-6t1dwhbsk"><div class="list-item-paragraph">length of the cone’s slant edge $l$,</div></li><li data-list-item-id="item-1778406103865-vlial8get" data-list-item-path="item-1778406103865-vlial8get"><div class="list-item-paragraph">total surface area of the container in contact with water.</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/1765252391651-nlzdrez95p.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNTIzOTE2NTEtbmx6ZHJlejk1cC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTMwMTE2LCJleHAiOjE3OTM1MzA0MTZ9.YHi9zyr1qw9VZJXjyeCVUsrHfThQmrF5FdgRiNVBV8o" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

An open container is made by joining a cylinder of radius $5$ cm and a cone of height $12$ cm. The container is completely filled with water. If the height of the cylinder is $9$ cm, give your answers in terms of $\pi $, where applicable.
Find the

Volume and Surface Area of a Joined Cylinder–Cone Container

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDcxMzE1OTEtc21pZ2dnaGpiZy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTMwMTE2LCJleHAiOjE3OTM1MzA0MTZ9.JXLDmB8y5GTHqaldh-_UL7dp0-02Ls6BGWVonSiWFMo" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDY3Nzg4OTktN3AxcnZzdmRscG8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.dujmFMkv0kpoxaxeaziarVa7-VBNAhtRXcV4K5ThWAM" 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

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDYwMzgxNzUtaTVsZXVuNGFxNy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTMwMTE2LCJleHAiOjE3OTM1MzA0MTZ9.bafxQLJTq9U3MHdO5ZvmdrceEQVtqfFrr67-wFdfmkE" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDQxMjQwNzgtd3BhaDBkeXRxZmEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.Up2y5vBmgb3Ny8gE3do6rnyo3Wpy39F8GgGzLRWiLHc" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDM2NzY4NTUtMndvM2M5Mm53cncucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.DREJtGJN0Yqq6snOVCp-Paw4qLo7GYRRFx-R_rcQ5m0" 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

<p>Diagram $1$ shows a cylindrical solid with height $10$ cm and radius $7$ cm.<br>Diagram $2$ shows a rectangular solid with a square base of sides $10$ cm and height $x$ cm.<br>Given that the volumes of the two solids are the same, find</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-1777972828428-1ntv8hzqj"><div class="list-item-paragraph">the volume of the cylindrical solid,</div></li><li data-list-item-id="item-1777972828879-21l3n4jca"><div class="list-item-paragraph">the value of $x$,</div></li><li data-list-item-id="item-1777972829687-5m8p8uls8"><div class="list-item-paragraph">the total surface area of rectangular solid.</div></li></ol><p>(Use $\pi =\frac{22}{7}$)</p></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/1764403287316-1e32j4t1ogo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDMyODczMTYtMWUzMmo0dDFvZ28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.OZZsgLnW1fB7W1mXJLxroceneGQ8EGVn2lZEPDSdwag" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Diagram $1$ shows a cylindrical solid with height $10$ cm and radius $7$ cm.
Diagram $2$ shows a rectangular solid with a square base of sides $10$ cm and height $x$ cm.
Given that the volumes of the two solids are the same, find

Comparing a Cylinder and a Rectangular Solid

<p>A stack of ten $\$1$ coins forms a cylinder of base diameter $x$ cm and height $2.5$ 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-1777885643496-t446ih76i" data-list-item-path="item-1777885643496-t446ih76i"><div class="list-item-paragraph">If the volume of the stack is $11.9$ cm$^{3}$, find $x$.</div></li><li data-list-item-id="item-1777885644052-37e9y6po2" data-list-item-path="item-1777885644052-37e9y6po2"><div class="list-item-paragraph">Find the total surface area of the stack.</div></li><li data-list-item-id="item-1777885645025-57qpa0jtd" data-list-item-path="item-1777885645025-57qpa0jtd"><div class="list-item-paragraph">If six more of the identical $\$1$ coins are added to the stack, find the percentage increase of the total surface area of the stack.</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/1764399699672-8t21c3pspn.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzOTk2OTk2NzItOHQyMWMzcHNwbi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTMwMTE2LCJleHAiOjE3OTM1MzA0MTZ9.CmU6TP-V1FnNcav_HlhbrUe-2-8dizId21uWlUol7lk" style="max-width: min(120px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

A stack of ten $\$1$ coins forms a cylinder of base diameter $x$ cm and height $2.5$ cm.

Percentage Increase and Decrease

Cylinder Formed by a Stack of Coins

Percentages

<p>The volume of the cylinder shown in the figure is $82$ cm${^3}$. If its height is $7$ cm, find the radius.</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/1764399291785-94xotp65uiq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzOTkyOTE3ODUtOTR4b3RwNjV1aXEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.s7H8QQihF3rdDPJaLAO39ZJ0LuG_MlcAYQ1WER-Z0b4" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

The volume of the cylinder shown in the figure is $82$ cm${^3}$. If its height is $7$ cm, find the radius.

Finding the Radius of a Cylinder from Its Volume

<p>The diagram shows an open cylinder of base diameter $14.2$ cm and height $22$ cm, being $\frac{2}{3}$ filled with a liquid.</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 data-list-item-id="item-1777884674343-95xwzphsu" data-list-item-path="item-1777884674343-95xwzphsu"><div class="list-item-paragraph">the base area of the cylinder.</div></li><li data-list-item-id="item-1777884675131-zroqazwkw" data-list-item-path="item-1777884675131-zroqazwkw"><div class="list-item-paragraph">the volume of liquid in this container.</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/1764399092537-899sow5dm77.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzOTkwOTI1MzctODk5c293NWRtNzcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.yPUTz_QqR-1hJglzAA3zUIt9BJg5RVE9C7N7oafBdJw" style="max-width: min(150px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows an open cylinder of base diameter $14.2$ cm and height $22$ cm, being $\frac{2}{3}$ filled with a liquid.

Base Area and Liquid Volume in an Open Cylinder

<p>The figure above shows a closed cylinder. Taking $\pi $ to be $3.14$, calculate</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-1786179245333-tp8x79ytw" data-list-item-path="item-1786179245333-tp8x79ytw"><div class="list-item-paragraph"><p>its volume,</p></div></li><li data-list-item-id="item-1786179246288-q778pof7x" data-list-item-path="item-1786179246288-q778pof7x"><div class="list-item-paragraph"><p>its total surface area.</p></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/1764398857405-1xbbpy152if.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzOTg4NTc0MDUtMXhiYnB5MTUyaWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.FxSjV2hcveGOTSpA31KC--9dxkIMZq6YVItOqKe74rs" style="max-width: min(180px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The figure above shows a closed cylinder. Taking $\pi $ to be $3.14$, calculate

Volume and Surface Area of a Closed Cylinder

<p>The curved surface area of the solid cylinder shown is $4713$ cm${^2}$. Find the volume of the cylinder.</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/1764398586134-w8upun0tbwi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzOTg1ODYxMzQtdzh1cHVuMHRid2kucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.52v8wlOTP7VvU9TScXpH6vMtT5TLsLL4sdpBK4UD-8M" style="max-width: min(180px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

The curved surface area of the solid cylinder shown is $4713$ cm${^2}$. Find the volume of the cylinder.

Finding Unknowns in a Cylinder

<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 metal rod has a diameter of $4.2$ cm.</p><p>Given that the volume of the rod is $225$ cm${^3}$, determine the</p><ol class="abc-list"><li data-list-item-id="item-1777719292673-ogomk7s33"><div class="list-item-paragraph">length,</div></li><li data-list-item-id="item-1777719293414-z5iwxl1n4"><div class="list-item-paragraph">external surface area of the rod.</div></li></ol><p>(Take $\pi =3.142$)</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/1764398422718-2zv6eipk3o5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQzOTg0MjI3MTgtMnp2NmVpcGszbzUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.ah2nnuNx-2m-IfbEu0JtiApeEbZ8smeX4cisXsr8G7w" style="max-width: min(180px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Problem involving Volume and Surface Area of a Cylinder

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzc5NzcwNTA4NzYtcnIzODZuMXBveWQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUzMDExNiwiZXhwIjoxNzkzNTMwNDE2fQ.nd0XB0OX9z_b2mm9R-BOPcEC-c2YUGxi4uywEZjDsmU" style="max-width: min(280px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

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