Question

Eight times the sum of the areas of two circular faces of a cylinder is equal to the four times the curved surface area of the cylinder. It is given that the diameter of the cylinder is 4n, where n is an integer and height of the cylinder is 8 cm. Find the value of n.

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Answer 1

Let the radius of the given cylinder be 'r' cm and height be 'h' cm.

Given, height of cylinder = h = 8 cm

Now, area of circular face of cylinder = $\pi$ $r^2$

Therefore, area of two circular faces = $\pi$ $r^2$ + $\pi$$r^2$ = 2 $\pi$$r^2$

Also, curved surface area of cylinder = 2$\pi$ rh = 2 $\pi$r $\times$ 8 = 16$\pi$r

According to the given condition,
8 $\times$ area of two circular faces of cylinder = 4 $\times$ curved surface area of cylinder
8 $\times$ 2 $\pi$$r^2$ = 4 $\times$ 16$\pi$ r
$\Rightarrow$ $r^2$ = 4r
$\Rightarrow$ r = 4
So, r = 4 cm
Hence, diameter of the cylinder = 2 $\times$ 4 cm = 8 cm
$\Rightarrow$ 4n = 8
$\Rightarrow$ n = 2