Question

The radius of a cylindrical gift box is (3x+4) inches. The height of the gift box is three times the radius. What is the surface area of the cylinder? Write your answer as a polynomial in a standard form.

Answer 1

The required equation of the surface area of the cylinder is $(60\pi x^2+192\pi x+128\pi)$

Solution:

Let "r" be the radius of cylinder and "h" be the height.

Formula for surface area is:

$S= 2r^2\pi + 2r\pi h$

where $2r^2\pi$ is surface of two bases of cylinder and $2r\pi h$ is surface of mantle.

From the given, the height of the gift box is three times the radius, i.e $h=3r=3\times(3x+4)\rightarrow h=9x+12$

On substituting the given values in the formula we get,

$S = 2\times\pi[(3x+4)^2 + (9x+12)\times(3x+4)]$

$\Rightarrow S=2\pi[(3x^2)+2\times3x\times4+4^2]+[(9x\times3x)+12\times3x+36x+48]$

$\Rightarrow S=2\pi[9x^2+24x+16]+[21x^2+36x+36x+48]$

$\Rightarrow S=2\pi[9x^2+24x+16+21x^2+72x+48]\rightarrow2\pi[30x^2+96x+64]$

On multiplying $2\pi$ inside the brackets we get,

$\Rightarrow S=[60\pi x^2+192\pi x+128\pi]$

Hence, the required equation is $(60\pi x^2+192\pi x+128\pi)$