The difference between outside and inside surface
Question:

The difference between outside and inside surface areas of cylindrical metallic pipe 14 cm long is 44 m2. If the pipe is made of 99 cm3 of metal, find the outer and inner radii of the pipe.

Solution:

We have to find the outer and inner radius of a hollow pipe.

Radius of inner pipe be $\left(r_{1}\right)$

Radius of outer cylinder be $\left(r_{2}\right)$

Length of the cylinder $(h)=14 \mathrm{~cm}$

Difference between the outer and the inner surface area is $44 \mathrm{~cm}^{2}$

So,

$2 \pi h\left(r_{2}-r_{1}\right)=44$

$2\left(\frac{22}{7}\right)(14)\left(r_{2}-r_{1}\right)=44$

So,

$\left(r_{2}-r_{1}\right)=\frac{1}{2}$………..(1)

So, volume of metal used is $99 \mathrm{~cm}^{3}$, so,

$\pi h\left(r_{2}^{2}-r_{1}^{2}\right)=99$

$\left(\frac{22}{7}\right)(14)\left(r_{2}-r_{1}\right)\left(r_{2}+r_{1}\right)=99$

Use equation (1) in the above to get,

$\left(\frac{22}{7}\right)(14)\left(\frac{1}{2}\right)\left(r_{2}+r_{1}\right)=99$

Therefore,

$\left(r_{2}+r_{1}\right)=\frac{9}{2} \ldots \ldots$(2)

Solve equation (1) and (2) to get,

$r_{2}=\frac{5}{2} \mathrm{~cm}$

$r_{1}=2 \mathrm{~cm}$

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