Question

ellis is painting wooden fence - posts before putting them in his yard. they are each 8 feet tall and have a diameter of 1 foot. there are 12 fence - posts in all. how much paint will ellis need to paint all the surfaces of the 12 fence - posts? use 3.14 for π, and round your answer to the nearest hundredth. provide an explanation and proof for your answer to receive full credit.

Explanation:

Step1: Find the surface - area formula for a cylinder

The fence - posts are cylindrical. The surface - area formula of a cylinder is $SA = 2\pi r h+ 2\pi r^{2}$. Since we are painting the fence - posts and we don't paint the bottom (when they are put in the ground), we use the formula $SA=\pi dh+\pi r^{2}$, where $d$ is the diameter, $h$ is the height, and $r$ is the radius. Given $d = 1$ foot, so $r=\frac{d}{2}=0.5$ foot and $h = 8$ feet.

Step2: Calculate the surface - area of one fence - post

Substitute the values into the formula:
\[

$$\begin{align*} SA&=\pi\times1\times8+\pi\times(0.5)^{2}\\ &=8\pi+\frac{\pi}{4}\\ &=\frac{32\pi+\pi}{4}\\ &=\frac{33\pi}{4}\text{ square feet} \end{align*}$$

\]

Step3: Calculate the total surface - area of 12 fence - posts

Multiply the surface - area of one fence - post by 12.
\[

$$\begin{align*} Total\ SA&=12\times\frac{33\pi}{4}\\ &= 99\pi\text{ square feet} \end{align*}$$

\]

Step4: Substitute $\pi = 3.14$ and round to the nearest hundredth

\[

$$\begin{align*} Total\ SA&=99\times3.14\\ &=310.86\text{ square feet} \end{align*}$$

\]

Answer:

310.86 square feet