A piece of wire of length 63 is cut, and the resulting tow pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the circle and the square?(a) Let x be the amount of wire used for the circle. What is the function A, the combined ara of the circle and square in terms of x?A= ______ (type an expression, type the exact number using pi as needed)To minimize the combined area, the wire should be cut so that a length of ____ is used for the circle and a length of ____ is used for the square. (round to nearest thousandth)(b) To maximize the combined area, the wire should be cut so that a length of ____ is used for the circle and a length of ____ is used for the square. (round to nearest thousandth)

Question
Asked Oct 29, 2019

A piece of wire of length 63 is cut, and the resulting tow pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the circle and the square?

(a) Let x be the amount of wire used for the circle. What is the function A, the combined ara of the circle and square in terms of x?

A= ______ (type an expression, type the exact number using pi as needed)

To minimize the combined area, the wire should be cut so that a length of ____ is used for the circle and a length of ____ is used for the square. (round to nearest thousandth)

(b) To maximize the combined area, the wire should be cut so that a length of ____ is used for the circle and a length of ____ is used for the square. (round to nearest thousandth)

check_circleExpert Solution
Step 1

The total length of the wire is 63.

Let x be the length of the wire cut for the circle.

Then, 63-x is the length of the ire cut for the square.

If x is the length cut for the circle then the circumference of the circle is x.

That is,

2πr
x
27T
-
x
The area of the circle is Tr
27T
47T
If 63-x is the length cut for the square then the perimeter of the square is 63-x.
That is, 4s 63-x
63 — х
63
Then, s
4
4
4
2
63
63x
3969
x
The area of the square is, s'
4
16
help_outline

Image Transcriptionclose

2πr x 27T - x The area of the circle is Tr 27T 47T If 63-x is the length cut for the square then the perimeter of the square is 63-x. That is, 4s 63-x 63 — х 63 Then, s 4 4 4 2 63 63x 3969 x The area of the square is, s' 4 16

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Step 2

Total area of the circle and the square is,

х* 63х 3969
A(x)
4л
16
8
16
63
+0
х
х
— 4 (к) -,
2л
8
63
х
х
2л
о
о
help_outline

Image Transcriptionclose

х* 63х 3969 A(x) 4л 16 8 16 63 +0 х х — 4 (к) -, 2л 8 63 х х 2л о о

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Step 3

Compute the maximum or mi...

A'(x)0
63
x
0 =
2π 8
8
63
2π8
47T
63
877
8
4 T
63
x
877
637
x
4 TT
x 27.71
help_outline

Image Transcriptionclose

A'(x)0 63 x 0 = 2π 8 8 63 2π8 47T 63 877 8 4 T 63 x 877 637 x 4 TT x 27.71

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