Chapter 8.3, Problem 30P

The radius of a cylindrical tennis ball can is $(2x+1)\text{cm}.$

The height of the tennis ball can is six times the radius. What is the surface area of the cylinder? Write your answer in simplified form.

To determine

To find: the surface area of the cylinder.

Answer to Problem 30P

Surface area of the cylinder is $\left(48\pi x^3+72\pi x^2+36\pi x+6\pi \right)$ square cm.

Explanation of Solution

Given information: The radius of a cylindrical tennis ball can is $\left(2x+1\right)$ cm. The height of the tin is six times the radius.

Calculation:

Let r and h be the radius and height of the cylindrical popcorn tin respectively.

Sine height of the tin is three times the radius, therefore h=6r

The formula for the surface area of the cylinder is

  $A=\pi r^{2}h$

Since, h=6r, therefore surface area of the tin is

  $\begin{array}{l}A=\pi r^2\left(6r\right)\\ A=6\pi r^3\end{array}$

Substituting the value $r=2x+1$

  $A=6\pi {\left(2x+1\right)}^3$

Now, use the identity, ${\left(a+b\right)}^3=a^3+3a^{2}b+3a{b}^2+b^3$

  $\begin{array}{l}A=6\pi \left[{\left(2x\right)}^3+3{\left(2x\right)}^2+3\left(2x\right)+1\right]\\ A=6\pi \left(8x^3+12x^2+6x+1\right)\\ A=48\pi x^3+72\pi x^2+36\pi x+6\pi \end{array}$

Thus, surface area of the cylinder is $\left(48\pi x^3+72\pi x^2+36\pi x+6\pi \right)$ square cm.