Question

The difference between outside and inside surface areas of cylindrical metallic pipe of 14 cm long is 44 $c{m^2}$. If the pipe is made of 99 $c{m^3}$ of metal, find the outer and inner radii of the pipe.

Answer 1

Hint: Here we will find the radii of outer and inner radii of pipe using the area and volume of cylindrical formula.

Complete step-by-step answer:
Let inner and outer radii of the cylindrical pipe be r and R respectively and height h=14 cm.
The difference between outside and inside surface area of cylindrical metallic pipe is
$2\pi (R - r) \times h = 44 \Rightarrow R - r = \dfrac{{44 \times 7}}{{44 \times 14}} = 0.5 \\\ \Rightarrow R - r = 0.5 \to (1) \\\$
Volume of pipe is

$$\Rightarrow \pi ({R^2} - {r^2}) \times h = 99 \\\ \Rightarrow {R^2} - {r^2} = \dfrac{{99 \times 7}}{{22 \times 14}} = 2.25 \\\ \Rightarrow (R + r)(R - r) = 2.25 \Rightarrow 0.5 \times (R + r) = 2.25 \\\ \Rightarrow R + r = 4.5 \to (2) \\\$$

Adding (1) and (2) we get
$R - r = 0.5 \\\ R + r = 4.5 \\\ 2R = 5 \Rightarrow R = 2.5 \\\$
Put value of R into equation (2) to get value of r
$r = 4.5 - 2.5 = 2$
The value of inner(r) and outer (R) radii are 2cm and 2.5 cm respectively.

Note: By applying the formula of surface area and volume of cylinder, we can easily solve questions. Mistakes can be avoided in the assignment of units.