IMO Practice Test — Mensuration
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
medium
A solid cone of height 24 cm and base radius 6 cm is sliced into two pieces by a plane parallel to the base at a height of 12 cm from the base. Find the ratio of the volume of the small top cone to the volume of the lower frustum.
1:7
1:8
1:3
1:4
Explanation: Scale factor between top cone and full cone is $12/24 = 1/2$. Vol ratio = $(1/2)^3 = 1/8$. Frustum vol is $8 - 1 = 7$ units. Ratio is 1:7.
Question 2 of 6
medium
Rainwater running off a flat rectangular roof of size 22 m $\cdot$ 20 m drains completely into a cylindrical tank of base radius 2 m. If a rainstorm causes the water level in the tank to rise by 3.5 meters, find the rainfall depth on the roof.
5 cm
8 cm
10 cm
12 cm
Explanation: Volume of water $= \pi \cdot 2^{2} \cdot 3.5 = \frac{22}{7} \cdot 4 \cdot 3.5 = 44$ cubic m. Roof area $= 22 \cdot 20 = 440$ sq. m. Rainfall depth $= \frac{44}{440} = 0.1$ m $= 10$ cm.
Question 3 of 6
medium
A solid metallic sphere of radius $R$ is melted down and cast into $n$ identical small solid spheres of radius $r$. Find the total surface area of all $n$ small spheres combined, expressed in terms of the original sphere's surface area $S$.
$n \cdot S$
$n^{1/3} \cdot S$
$n^{2/3} \cdot S$
$S / n$
Explanation: $V = n \cdot v \implies R^3 = n \cdot r^3 \implies r = R / n^{1/3}$. Total area = $n \cdot (4\pi r^2) = n \cdot 4\pi(R^2 / n^{2/3}) = n^{1/3} \cdot 4\pi R^2 = n^{1/3}S$.
Question 4 of 6
medium
A cylindrical bucket 12 cm tall with a base radius of 8 cm is filled completely with sand. This bucket is emptied onto flat ground to form a perfect conical heap of sand. If the height of the conical heap is 16 cm, find its slant height.
9 cm
12 cm
15 cm
18 cm
Explanation: Volume is conserved: $\pi \cdot 8^{2} \cdot 12 = \frac{1}{3}\pi r^{2} \cdot 16$, so $768 = \frac{16}{3}r^{2} \Rightarrow r^{2} = 144 \Rightarrow r = 12$. Slant height $l = \sqrt{r^{2}+h^{2}} = \sqrt{12^{2}+16^{2}} = \sqrt{144+256} = \sqrt{400} = 20$ cm.
Question 5 of 6
medium
A hemi-spherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into small cylindrical bottles of diameter 3 cm and height 4 cm. How many bottles are needed?
24
36
48
54
Explanation: Vol bowl = $(2/3)\pi \cdot 9^3 = 486\pi$. Vol bottle = $\pi \cdot 1.5^2 \cdot 4 = 9\pi$. Number of bottles $n = 486\pi / 9\pi = 54$.
Question 6 of 6
medium
A solid right circular cone has a volume of $V_1$. A second solid right circular cylinder has a volume of $V_2$. If the cylinder has twice the radius and half the height of the cone, find the exact ratio $V_1 : V_2$.
1:6
2:3
3:2
6:1
Explanation: $V_1 = (1/3)\pi r^2h$. $V_2 = \pi(2r)^2(h/2) = \pi \cdot 4r^2 \cdot (h/2) = 2\pi r^2h$. Ratio $V_1 / V_2 = ((1/3)\pi r^2h) / (2\pi r^2h) = 1/6$.