A piece of wire 24 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much wire should be used for the square in order to maximize the total area? Incorrect: Your answer is incorrect. m (b) How much wire should be used for the square in order to minimize the total area?

A) 23 m; B) 12 m

Step-by-step explanation:

The larger the circle, the larger the total area will be.

This means we want to use as much wire as possible for the circle. If we use only 1 meter for the rectangle, this means each side would be 0.25 m; this makes the area of the rectangle 0.25 (0.25) = 0.0625 m².

This leaves us 24-1 = 23 m of wire for the circle. This will be the distance around the circle, or the circumference. The formula for circumference is

C = 2πr; using 23 for C and 3.14 for π, we have

23 = 2 (3.14) r

23 = 6.28r

Divide both sides by 6.28:

23/6.28 = 6.28r/6.28

3.66 = r

The formula for area is A = πr²; using 3.14 for π and 3.66 for r, we have

A = 3.14 (3.66) ² = 42.06 m².

This makes the total area 42.06+0.0625 = 42.1225 m²

B) The closer the amount of wire is to half of the total, the smaller the total area will be.

If we cut the wire exactly in half, there will be 12 m for the rectangle; this means each side will be 12/4 = 3 m long. This makes the area 3 (3) = 9 m².

This leaves us with 24-12 = 12 m for the circle. This is the circumference; using 3.14 for π, we have

12 = 2 (3.14) r

12 = 6.28r

Divide both sides by 6.28:

12/6.28 = 6.28r/6.28

1.91 = r

This makes the area

A = πr² = 3.14 (1.91) ² = 11.46

This gives us a total area of 11.46+9 = 20.46 m²