Question

Steve likes to entertain friends at parties with "wire tricks." Suppose he takes a piece of wire 68 inches long and cuts it into two pieces. Steve takes the first piece of wire and bends it into the shape of a perfect circle. He then proceeds to bend the second piece of wire into the shape of a perfect square. What should the lengths of the wires be so that the total area of the circle and square combined is as small as possible? (Round your answers to two decimal places.)
I have attempted it several times and my answers add up to 68 but are wrong!

Answer 1

I am getting 29.91 inches for the circle wire and 38.09 inches for the square wire.
If it is correct then I can type my steps here. If not I can just delete this answer.

Answer 2

That is correct how did you find those numbers

Answer 3

The wire is 68 inches long.
Assume C is the length of the wire cut for the circle.
Then the length of the wire remaining for the square is (68 - C)

C is the length of the circumference of a circle.
So 2(pi)R = C, R = C/2pi
Area of a circle = (pi)R^2 = (pi)C^2/(4pi^2) = C^2/(4pi)

(cont'd)

Answer 4

The length of the wire for the square is (68 - C)
The square has 4 equal sides adding up to (68 - C)
So the side of the square must be (68-C)/4
The area of the square = (68-C)^2 / 16

Total area of circle + square = C^2/(4pi) + (68-C)^2 / 16

Answer 5

then what do you do with the equations?

Answer 6

Let me put in values for pi and simplify the numbers

Total Area = C^2/(4 x 3.1416) + (4624 + C^2 - 136C) / 16
= C^2 / 12.5664 + 289 + 0.0625C^2 - 8.5C
= C^2 (1/12.5664 + 0.0625) - 8.5C + 289
= 0.1421C^2 - 8.5C + 289 = 0

To find the maximum/minimum, find the derivative with respect to C and set it to zero:

dA/dC = 0.2842C - 8.5 = 0

C = 29.91 inches

Answer 7

So the wire length for the circle is 29.91 inches and the wire remaining for the square is (68 - 29.91) = 38.09 inches

The second derivative is 0.2842 which is positive and therefore we have found a minimum

Wire length for circle = 29.91 inches
Wire length for square = 38.09 inches

Answer 8

ok so then to find the total minimum area you plug the values back in

Answer 9

Correct. Total area A = 0.1421C^2 - 8.5C + 289
Put C = 29.91 and calculate A.

Answer 10

Total Area A = 161.89 square inches.

If you want individual areas of the circle and the square, put the values in the individual equations for the area.

Answer 11

I tried it and it did not provide the correct answer? Let me try the individual answers

Answer 12

The question you posted above only asks for the lengths.

Answer 13

The second half is what is the combined minimal area.

Answer 14

It accepted the two lengths as correct but not 161.89 for the combined minimal area?

Answer 15

Yes the lengths were right but 161.89 did not work

Answer 16

DO they tell you to what decimal place to round off the area?

Answer 17

two decimals

Answer 18

Without any of the above formula you can recalculate the area of a circle formed by a wire 29.91 inches long and the area of a square formed by a wire of length 38.09 inches and you will get the area to be 161.89.

Answer 19

Yea I have tried recalculating it and I get the same answer however it tells me its not correct. It is alright thank you for your help with finding the lengths

Answer 20

Try 161.87 It may be the second decimal that it is complaining about.

Answer 21

It still didn't like it

Answer 22

Oh, well. But if you keep trying different second decimal it might eventually click. Doesn't make any sense. But anyway, you are welcome.