Geometry

Ellis is painting wooden fenceposts before putting them in his yard. They are each 7 feet tall and have a diameter of 1 foot. There are 11 fenceposts in all. How much paint will Ellis need to paint all the surfaces of the 11 fenceposts? 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.

Step-by-step solution with explanation

Final Answer

Ellis needs of paint to cover all surfaces of the 11 fenceposts.

Step-by-step solution

1

Identify the shape and surfaces

Each fencepost is a cylinder. We need to paint ALL surfaces: the long curved side (lateral surface) and both flat circular ends (top and bottom).
2

Find the radius from the diameter

The diameter is 1 foot, so the radius is half of that, which is 0.5 feet. We need the radius for the area formulas.
3

Calculate the lateral surface area

The lateral (side) surface area of a cylinder is . We multiply 2 × 3.14 × 0.5 × 7 to get 21.98 square feet for one post's curved side.
4

Calculate the area of both circular bases

Each circular end has area , and there are two ends per post. So we multiply by 2: square feet.
5

Find total surface area of one fencepost

Add the lateral surface area and the two bases together. One fencepost has a total surface area of 23.55 square feet.
6

Multiply by 11 for all fenceposts

There are 11 identical fenceposts. Multiply the surface area of one post by 11 to get the total area Ellis needs to paint.

Understanding this problem

Learning Insight

A cylinder's surface area comes from 'unrolling' the curved side into a flat rectangle. That rectangle's dimensions are the height of the cylinder and the circumference of the circle (). This is why the lateral formula looks like circumference times height. The two circular ends are just added on top of that.

Quick Tip

Remember the full surface area formula as . You can factor it as , which saves a step and reduces rounding errors when calculating by hand.

Common Mistake

Many students forget to include BOTH circular bases — they only add one base () instead of two (), or they forget the bases entirely and only calculate the lateral surface area.