Question

The curved surface area of a cylindrical pillar is $ 264 $ sq.cm and its volume is $ 924 $ cubic cm. The ratio of its diameter to its height is:
 $
  (a)\,\,3:7 \\
  (b)\,\,7:3 \\
  (c)\,\,6:7 \\
  (d)\,\,7:6 \\
  $

Answer 1

Hint: To find required ratio, we first form two equations one from surface area of cylinder and other from volume of cylinder and then solve these equations together to get the value of ‘r’ and ‘h’ of cylinder. Then from radius we find diameter and finally required ratio of diameter to its height.
Volume of cylinder = $ \pi {r^2}h, $ Curved surface area of cylinder = $ 2\pi rh $ , where ‘r’ and ‘h’ are radius and height of cylinder respectively.

Complete step-by-step answer:
Let the radius of the cylindrical pillar be r and height h.
We know,
Curved Surface Area of the cylinder = $ 2\pi rh $
Substituting values in above formula. We have,
 $
  264 = 2 \times \dfrac{{22}}{7} \times r \times h \\
   \Rightarrow r \times h = 264 \times \dfrac{7}{{22}} \times \dfrac{1}{2} \\
   \Rightarrow rh = 132 \times \dfrac{7}{{22}} \\
   \Rightarrow rh = 6 \times 7 \\
   \Rightarrow rh = 42.......................(i) \\
  $
Also, volume of cylinder is given as $ \pi {r^2}h $
 $ \Rightarrow V = \pi {r^2}h $
Substituting values in above formula. We have,
\[
  924 = \dfrac{{22}}{7} \times {r^2} \times h \\
   \Rightarrow {r^2} \times h = 924 \times \dfrac{7}{{22}} \\
   \Rightarrow {r^2}h = 42 \times 7 \\
   \Rightarrow {r^2}h = 294....................(ii) \\
 \]
Dividing (ii) by (i) we have
 $
  \dfrac{{{r^2}h}}{{rh}} = \dfrac{{294}}{{42}} \\
  \Rightarrow r = 7 \\
  $
Now, substituting the value of r in equation (i) to get value of h.
 $
  rh = 42 \\
   \Rightarrow 7(h) = 42 \\
   \Rightarrow h = \dfrac{{42}}{7} \\
   \Rightarrow h = 6 \\
  $
Therefore, from above we see that the radius of the cylinder is $ 7cm $ and height is $ 6\,cm. $
From above diameter of cylinder is = $ (2r) $
Diameter of cylinder is = $ 2 \times 7 = 14\,cm $
Now, to find the ratio of diameter to height. We have,
 $
  \dfrac{{diameter}}{{height}} = \dfrac{{14}}{6} \\
   \Rightarrow \dfrac{{diameter}}{{height}} = \dfrac{7}{3} \\
  $
Hence, from above we see that ratio of diameter to its height is $ 7:3\,. $
So, the correct answer is “Option B”.

Note: While doing mensuration problems one must see units related to terms given. If there are different units then the first step is to make their units the same and then form different equations as per the given condition and then solve them together to find the required solution of the problem.