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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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzE3NTcxNzAtOHF0Z3B4M3ViZ2UucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.bMvCFlJlU0gSPcjQZgOyFzj__KSH_sQplB7L-HeEDP0" 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>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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzE1OTE4NjItMmlxbGNvOHdrc2sucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.mPnfMsOtbjISjiNTrN8IwwCclPBY42VPBbrYqeOBkB8" 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.

Calculating Rates and Quantities

Ratio, Rate and Speed

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NzExNjI2NzQtZWpjZmRpYjVzMW8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.4wOZ7b420FqpJeEsGLHRtWYquz_tuBJVe6HaKi-OQPM" 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>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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTU2MjQ4NDcteHpia3RnMDJ1b2cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.0LEj_fXWXELByl7PJp0ayho7VpI4FTRkVuD7kNWIWGY" 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 Composite Solids

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQ5MjU3MjktZTNzZ3JnZndrOWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.g-DIt2hG7v9OLCVnJbdr4jNeNY7-rnsykAY8LYjaVgw" 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

Mensuration (Pyramid, Cones and Spheres)

Linear Equations and Inequalities

<p>The figure below shows a pentagonal prism. Given that the perimeter of its cross section is $489.4$mm, find, correcting your answers to the nearest integer,</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/1767694189025-w6omtkdpz0o.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTQxODkwMjUtdzZvbXRrZHB6MG8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.eX9pDGT3kr-UMxVPHQkiD2GHZJRaPf2ae6rGIZblCbg" style="max-width: min(400px, 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">the value of $x$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the area of the cross section in cm$^2$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total surface area of the prism in cm$^2$, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">the volume of the prism in cm$^3$.</span></div></li></ol>

The figure below shows a pentagonal prism. Given that the perimeter of its cross section is $489.4$mm, find, correcting your answers to the nearest integer,

the total surface area of the prism in cm$^2$, and

Conversion between Cubic Units

Conversions between Square Units

Perimeter and Area of Plane Figures

<p>The figure below shows a net of a prism, which consists of $3$
 pairs of identical rectangles.</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/1767693836575-2pvpkqevsea.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTM4MzY1NzUtMnB2cGtxZXZzZWEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.MbSC62fr63ytlSICgwqwNIrEYvSbf1bJSwsHJfNDJ9s" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">State the type of the prism.</div></li><li><div class="list-item-paragraph">Find its total surface area, and its volume</div></li></ol>

The figure below shows a net of a prism, which consists of $3$
 pairs of identical rectangles.

<p>The figure below shows a net of a prism, which consists of $6$ identical squares.</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/1767693685059-i1p07qgbaw.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTM2ODUwNTktaTFwMDdxZ2Jhdy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQxMTcxLCJleHAiOjE3OTM1NDE0NzF9.UltGVCeoYDTEtY85FLbeM-zuf1IBpIZxDlhq_J4lAQg" 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">State the type of the prism.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find its total surface area, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find its volume.</span></div></li></ol>

The figure below shows a net of a prism, which consists of $6$ identical squares.

Recognising Cube Nets

<p>The figure below shows a hexagonal prism, where the bases are regular hexagons.</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/1767693262807-h12zr7l0qp9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTMyNjI4MDctaDEyenI3bDBxcDkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.oMZti0ujJ48lpQwlQdkJmouuF9UoW3d2AzWvzTK0UsY" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">its total surface area, and</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">its volume.</div></li></ol>
<p>The 10.4 cm distance between opposite sides is a rounded diagram measurement. For this exercise use that supplied measurement; the resulting areas and volume are approximate. The exact distance for a regular hexagon of side 6 cm is $6\sqrt3\text{ cm}$.</p>

The figure below shows a hexagonal prism, where the bases are regular hexagons.

The 10.4 cm distance between opposite sides is a rounded diagram measurement. For this exercise use that supplied measurement; the resulting areas and volume are approximate. The exact distance for a regular hexagon of side 6 cm is $6\sqrt3\text{ cm}$.

<p>The figure below shows a trapezoidal prism that has a total surface area of 304.5 cm$^2$. </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/1767693125191-kgcrqbr5r3a.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTMxMjUxOTEta2djcnFicjVyM2EucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.8eqbostSEY5uAqw5PHv7tQ6Fp4E-_ZiIzqB0z86BFSU" 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">the value of $h$, and</span></div></li><li><div class="list-item-content"><span class="list-item-text">its volume.</span></div></li></ol>

The figure below shows a trapezoidal prism that has a total surface area of 304.5 cm$^2$. 

<p>The figure below shows a trapezoidal prism.</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/1767692975059-uq21v4otjq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTI5NzUwNTktdXEyMXY0b3RqcS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQxMTcxLCJleHAiOjE3OTM1NDE0NzF9.B6A5zakS9MYttsjSkwvilvymBUGeOhumnFqYI0MDT78" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">its total surface area, and</div></li><li><div class="list-item-paragraph">its volume.</div></li></ol>

The figure below shows a trapezoidal prism.

<p>The figure below shows a triangular prism that has a volume of $972$
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/1767692842255-nyhk3ptqaj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTI4NDIyNTUtbnloazNwdHFhai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQxMTcxLCJleHAiOjE3OTM1NDE0NzF9.Z2ooBtvyV4qlrcaGiHbaE2aBOO41YLT4wbh2tdJ5C5s" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">the value of $h$, and</div></li><li><div class="list-item-paragraph">its total surface area.</div></li></ol>

The figure below shows a triangular prism that has a volume of $972$
cm$^3$
.

<p>The figure below shows a triangular prism. </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/1767692529672-cxsvwpis8z.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTI1Mjk2NzItY3hzdndwaXM4ei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQxMTcxLCJleHAiOjE3OTM1NDE0NzF9.FEhYrHmYphJwkLXs9O00Ld_aWo8XvGvBqCMasDiZ3N8" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">its total surface area, and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">its volume.</div></li></ol>
<p>Correction to the source diagram: take the three triangle side lengths 12 cm, 12 cm and 7.5 cm as exact. The printed perpendicular height 7.2 cm is inconsistent with these lengths. Calculate the height from the side lengths instead.</p>

The figure below shows a triangular prism. 

Correction to the source diagram: take the three triangle side lengths 12 cm, 12 cm and 7.5 cm as exact. The printed perpendicular height 7.2 cm is inconsistent with these lengths. Calculate the height from the side lengths instead.

<p>A cuboid has dimensions of $14$ cm by $4$ cm by $4.5$ cm.</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/1767692407709-16tgzrrtipu.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2OTI0MDc3MDktMTZ0Z3pycnRpcHUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.XxHCLUdX8NiWp3xGc7Hz2ziyF3HCFIF_sCUrVHNCwoE" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="i-list"><li data-list-item-id="item-1767692416171-35v5jsi1e" data-list-item-path="item-1767692416171-35v5jsi1e"><div class="list-item-paragraph">its total surface area, and</div></li><li data-list-item-id="item-1767692417139-4ay2qpzu0" data-list-item-path="item-1767692417139-4ay2qpzu0"><div class="list-item-paragraph">its volume.</div></li></ol>

A cuboid has dimensions of $14$ cm by $4$ cm by $4.5$ cm.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjE5NzgxMjItMzdxcTNjeTkwOTYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ._Oy7-oMJzoUPyW31HcI7IkMESoBEdtXiwng98F9oZPs" 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 Transfer Between Composite Solids

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUyNjA1NjA0ODctazY5ZWJhY3d5cGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.T2CiUI5u6HYdPSUxrLvgKnXP5vAxkuj0upKGySyqtUw" 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

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDY0NzUwNDYtcXYzaWpsaHhpY2cuSlBHIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.V7QTAfUI4b5szdsl_MraKcSmBBagNB9gCVGFhzsgNKw" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDYwMzgxNzUtaTVsZXVuNGFxNy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQxMTcxLCJleHAiOjE3OTM1NDE0NzF9.W-WIXw0e3Cxboey65-SUyTiqNPz-W2ZQCaR2NgAm_ww" 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>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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDMyODczMTYtMWUzMmo0dDFvZ28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU0MTE3MSwiZXhwIjoxNzkzNTQxNDcxfQ.hk4GB15urt-Z8rqWN85OOd3kRI9r77JB_q3DhlzOgRI" 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 prism is shown in the diagram.  The volume of the prism is $567$ cm$^{3}$. Find the height, $h$, of the prism.</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/1764402918944-kskqzp3ucq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjQ0MDI5MTg5NDQta3NrcXpwM3VjcS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTQxMTcxLCJleHAiOjE3OTM1NDE0NzF9.nGwNprWAjdIhC_E9jw7gIHtSieOkEvKJNB3hQRPtKdc" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

A prism is shown in the diagram.  The volume of the prism is $567$ cm$^{3}$. Find the height, $h$, of the prism.

Math Question Bank