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CAT 2017 Slot 1 — QA Question 19

SphereEasy

A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is 3 cm, while its volume is 9π cm3. Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is:

Answer & solution

Correct answer: 6

Answer: 6

Solution

Easy

The ball (radius 22) rests on the rim of the open top, so it dips inside until its surface touches the rim circle (radius rr). The centre sits a height aa above the rim, where aa, rr and the ball-radius form a right triangle. Total height == cylinder + a ++\,a\,+ ball radius.

O R=2 a h=3
1

Cylinder radius from its volume. Height h=3h=3, volume 9π9\pi:

πr2h=9π⇒ πr2(3)=9π⇒ r2=3⇒r=3\begin{aligned} &\pi r^2 h=9\pi\\ &\Rightarrow\ \pi r^2(3)=9\pi\\ &\Rightarrow\ r^2=3\Rightarrow r=\sqrt{3} \end{aligned}
2

Height of the ball's centre above the rim. Ball radius R=42=2R=\tfrac{4}{2}=2. The rim point, centre OO, and the dip aa form a right triangle with hypotenuse RR:

R2=a2+r2⇒ 22=a2+(3)2⇒ a2=4−3=1⇒a=1\begin{aligned} &R^2=a^2+r^2\\ &\Rightarrow\ 2^2=a^2+(\sqrt3)^2\\ &\Rightarrow\ a^2=4-3=1\Rightarrow a=1 \end{aligned}
3

Add up the vertical distances. Base to rim ++ rim to centre ++ centre to top:

height=h+a+R=3+1+2=6\begin{aligned} &\text{height}=h+a+R=3+1+2=6 \end{aligned}
Topmost height=6 cm\text{Topmost height}=6\ \text{cm}

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CAT 2017 Slot 1 QA Q19: A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is — Solution | TheCATExam