Question 7
A sphere and a right circular cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height?

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Solution

Let the radius of the sphere = r = Radius of a right circular cylinder
According to the question,
Volume of cylinder = Volume of a sphere
$\Rightarrow π r^{2} h = \frac{4}{3} π r^{3} \Rightarrow h = \frac{4}{3} r$
$∵$ Diameter of the cylinder = 2r
$∴$ Difference between diameter and height of the cylinder
$= 2 r - \frac{4 r}{3} = \frac{2 r}{3}$

Now, percentage increase in diameter of the cylinder compared to its height

$= \frac{i n c r e a s e i n d i a m e t e r o f t h e c y l i n d e r c o m p a r e d t o i t s h e i g h t \times 100}{h e i g h t o f t h e c y l i n d e r}$

$= \frac{\frac{2 r}{3} \times 100}{\frac{4}{3} r} = 50 %$

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