A florist uses wire frames to support flower arrangements displayed at weddings. Each frame is constructed from a wire length 9ft long that is cut into six pieces. the vertical edges of the frame consist of four of the pieces of wire that are each 12 inches long. one of the remaining pieces is bent into a square to form the base of the frame; the final piece is bent into a circle to form the top of the frame. how should the florist cut the wire of the length 9ft in order to minimize the combined area of the circular top and square base of the frame?

Question

A florist uses wire frames to support flower arrangements displayed at weddings. Each frame is constructed from a wire length 9ft long that is cut into six pieces. the vertical edges of the frame consist of four of the pieces of wire that are each 12 inches long. one of the remaining pieces is bent into a square to form the base of the frame; the final piece is bent into a circle to form the top of the frame. how should the florist cut the wire of the length 9ft in order to minimize the combined area of the circular top and square base of the frame?

I'm not really sure what this is even asking.

Answer 1

I made a diagram according to you description and concluded the following

Since each of the 4 "verticals" is 12 inches, that would leave 5 feet or 60 inches for the circle and the square base
Let the radius of the circle be x feet
then the length for the base would be 5 - 2πx feet
and each side is 5/4 - (1/2)πx

let the total area of top and bottom be A

A = πx^2 + (5/4 - (1/2)πx)^2

differentiate, set that equal to zero and solve for x
I will let you finish the messy part.

Answer 2

To minimize the combined area of the circular top and square base of the frame, we need to determine the dimensions of the square and circle that would use up the least amount of wire.

Let's break down the information given:

The total wire length is 9ft, which is equivalent to 108 inches.

The vertical edges of the frame consist of four pieces of wire, each 12 inches long. So, the total length used for the vertical edges is 4 * 12 inches = 48 inches.

We are left with two pieces of wire, one for the square base and the other for the circular top.

To minimize the combined area, we should aim to make both the square and circle as small as possible.

Let's start with the square base:

Since a square has all four sides equal in length, we can divide the remaining wire equally, using 24 inches for each side of the square.

Next, let's move on to the circular top:

A circle's circumference is calculated with the formula C = 2πr, where r is the radius.

We have 60 inches (108 inches - 48 inches - 24 inches) remaining for the circle.

To minimize the circumference, we need to minimize the radius, as the circumference is directly proportional to the radius. Therefore, we want to use the maximum possible length for the radius.

Using the formula for circumference mentioned earlier, we can determine the maximum radius we can achieve given the remaining wire.

60 inches = 2πr

Dividing both sides by 2π, we get:

r = 60 inches / (2π)

r ≈ 9.55 inches

We need to keep in mind that we are working with pieces of wire, so we cannot have a radius of 9.55 inches exactly. Therefore, to minimize the overall area, we would choose a radius of 9 inches instead (assuming we can cut to the nearest inch).

To summarize:

1. Cut four pieces of wire, each 12 inches long, for the vertical edges.
2. Cut one piece of wire, 24 inches long, to form the square base.
3. Cut one piece of wire, 60 inches long, and approximate the radius of the circle to 9 inches.

By following these cutting instructions, the florist can minimize the combined area of the circular top and square base of the frame using a wire length of 9ft.