Question

ellis is painting wooden fenceposts before putting them in his yard. they are each 6 feet tall and have a diameter of 1 foot. there are 12 fenceposts in all. how much paint will ellis need to paint all the surfaces of the 12 fenceposts? use 3.14 for \\(\pi\\), and round your answer to the nearest hundredth. provide an explanation and proof for your answer to receive full credit.

Explanation:

Identify given dimensions

We analyze the shape of a single fencepost.
A fencepost is modeled as a cylinder.
Height \(h = 6\text{ ft}\).
Diameter \(d = 1\text{ ft}\).
Radius \(r = \frac{d}{2} = 0.5\text{ ft}\).
Number of fenceposts \(N = 12\).
Value of \(\pi \approx 3.14\).

Calculate single post surface area

We find the total surface area of one cylinder.
The formula for total surface area is:

$$A_{\text{single}} = 2\pi r^2 + 2\pi r h$$

Substitute the given values:

$$A_{\text{single}} = 2(3.14)(0.5)^2 + 2(3.14)(0.5)(6)$$

$$A_{\text{single}} = 2(3.14)(0.25) + 2(3.14)(3)$$

$$A_{\text{single}} = 1.57 + 18.84 = 20.41\text{ sq ft}$$

Calculate total surface area

We multiply by the total number of posts.
There are 12 fenceposts in total:

$$A_{\text{total}} = N \times A_{\text{single}}$$

$$A_{\text{total}} = 12 \times 20.41$$

$$A_{\text{total}} = 244.92\text{ sq ft}$$

Answer:

Ellis will need enough paint to cover \(244.92\text{ square feet}\) of surface area.