Question

The difference between the outer and the inner curved surfaces areas of an open cylinder is $88{\text{ }}c{m^2}$. If its length is 14 cm and volume of the material in it is $176{\text{ }}c{m^3}$, find the inner diameter of the cylinder.
A) 3 cm
B) 4 cm
C) 2 cm
D) 1 cm

Answer 1

Hint:
Apply the formula to find the difference between the outer and the inner curved surfaces areas of a cylinder and equate it to the given values. Similarly, apply the formula to find the difference between the outer and the inner volumes and equate it to $176{\text{ }}c{m^3}$. When you get two equations, solve them simultaneously to find the required value.

Complete step by step solution:
Let us begin with the formula to find the difference between the outer and the inner curved surfaces areas of a cylinder, which is given by;
$2\pi h\left( {R - r} \right)$, where ‘R’ is the radius of the outer curved surface and ‘r’ is the radius of the inner curved surface.
It is given in the question that this difference is $88{\text{ }}c{m^2}$.
$
   \Rightarrow 2\pi h\left( {R - r} \right) = 88 \\
   \Rightarrow R - r = \dfrac{{88 \times 7}}{{2 \times 22 \times 14}} \\
   \Rightarrow R - r = 1{\text{ ......(1)}} \\
 $
Next, we find the difference between the outer and the inner volumes, using the formula, $V = \pi h\left( {{R^2} - {r^2}} \right)$ . It is given in the question that this difference is$176{\text{ }}c{m^3}$.
$
   \Rightarrow \pi h\left( {{R^2} - {r^2}} \right) = 176 \\
   \Rightarrow \left( {\dfrac{{22}}{7}} \right)\left( {14} \right)\left( {{R^2} - {r^2}} \right) = 176 \\
   \Rightarrow \left( {R - r} \right)\left( {R + r} \right) = \dfrac{{176 \times 7}}{{22 \times 14}} \\
   \Rightarrow \left( 1 \right)\left( {R + r} \right) = 4{\text{ (from (1))}} \\
   \Rightarrow R + r = 4{\text{ ......(2)}} \\
 $
Now we will add the equations (1) and (2) to find the value of R.
$
   \Rightarrow R - r + R + r = 1 + 4 \\
   \Rightarrow 2R = 5 \\
   \Rightarrow R = \dfrac{5}{2} \\
 $
Now we know that the diameter is the double of the radius.
$
   \Rightarrow D = 2R \\
   \Rightarrow D = 2\left( {\dfrac{5}{2}} \right) \\
   \Rightarrow D = 5 \\
 $
This is the outer diameter of the cylinder.
Now, put the value of R in equation (2) to find the value of ‘r’.
$
   \Rightarrow R + r = 4 \\
   \Rightarrow \dfrac{5}{2} + r = 4 \\
   \Rightarrow r = 4 - \dfrac{5}{2} \\
   \Rightarrow r = \dfrac{3}{2} \\
 $
Now we know that the diameter is the double of the radius.
$
   \Rightarrow d = 2r \\
   \Rightarrow d = 2\left( {\dfrac{3}{2}} \right) \\
   \Rightarrow d = 3 \\
 $
This is the inner diameter of the cylinder.

Hence, option (A) is the correct option.

Note:
While allotting any variable to the radius, diameter follows a pattern in order to keep a track of calculation. Remember the basic formulas like the difference between the outer and the inner curved surfaces areas of a cylinder is $2\pi h\left( {R - r} \right)$. The diameter is twice its radius.