EduNinja

[Maximum number: 9]

(a)

A cylinder with radius 6 cm and height $h \mathrm{~cm}$ has the same volume as a sphere with radius 4.5 cm. Find the value of h.
[The volume, V, of a sphere with radius r is $V=\frac{4}{3} \pi r^{3}$ .]

h=

[ 3 ]

3.375 cao
M2 for $\frac{\frac{4}{3}\pi(4.5)^3}{\pi6^2}$ , or M1 for a correct volume equation.

(b)

A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius $r \mathrm{~cm}$ .

Find the value of r.
[The volume, V, of a sphere with radius r is $V=\frac{4}{3} \pi r^{3}$ .]

r=

[ 3 ]

3.63 or 3.627 to 3.628
M2 for $\frac{20^3}{40\times\frac{4}{3}\pi}$ , or M1 for a correct equation.

(c)

A solid cylinder has radius $x \mathrm{~cm}$ and height $\frac{7 x}{2} \mathrm{~cm}$ .

The surface area of a sphere with radius $R \mathrm{~cm}$ is equal to the total surface area of the cylinder.
Find an expression for R in terms of x.
[The surface area, A, of a sphere with radius r is $A=4 \pi r^{2}$ .]

R=

[ 3 ]

$\frac{3x}{2}$ or 1.5x or $1\frac{1}{2}x$
B2 for $4R^2=9x^2$ , or M1 for a correct surface-area equation.

[Maximum number: 7]

The diagram shows a prism, ABCDEF.
$A B=13 \mathrm{~cm}, A C=20 \mathrm{~cm}, C F=24 \mathrm{~cm}$ and angle $A B C=90^{\circ}$ .

(a)

Calculate the total surface area of the prism.
$\mathrm{cm}^{2}$

[ 6 ]

1350 or 1354....

6

M2 for $20^{2}-13^{2}$ or M1 for $B C^{2}+13^{2}=20^{2}$

A1 for $\sqrt{231}$ or 15.2 or 15.19 to 15.20

M1 for $20 \times 24$ and $13 \times 24$ and their

$15.2 \times 24$

M1 for $\left[\frac{1}{2} \times\right]$ their $15.2 \times 13$

(b)

Calculate the volume of the prism. $\mathrm{cm}^{3}$

[ 1 ]

2370 or 2369 to $2371 \ldots$ cao

1

[Maximum number: 6]

NOT TO SCALE

The diagram shows a sector OPQ of a circle with centre O and radius 24 cm. The sector angle is $60^{\circ}$ .

A cone is made from this sector by joining OP to OQ.

NOT TO SCALE

Calculate the volume of the cone.
[The volume, V, of a cone with radius r and height h is $V=\frac{1}{3} \pi r^{2} h$ .]

396 or 397 or 396.4 to 396.6

M2 for $[r]=(\frac{60}{360}\times2\pi\times24)\div2\pi$ oe or better
or M1 for $2\pi r=\frac{60}{360}\times2\pi\times24$ oe
M2 for $\sqrt{24^2-a^2}$ or M1 for $h^2+a^2=24^2$
M1 for $\frac13\pi\times\text{their }r^2\times\text{their }h$

CAIE IGCSE Mathematics Extended E5.4 Surface Area and Volume Topic Practice

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question 3