Question

PLEASE HELP!!!!! A wire 340 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?

Answer 1

Answer:
The piece of wire cut for the square is 180.3 inches, and the piece of wire cut for the circle is 159.7 inches.
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Explanation:
Use the following formulas for the areas of a square and circle:

Square
$A = s^{2}$

Circle
$A = \pi r^{2}$

s represents the length of the side in a square, and r represents the radius of the circle.

Set these two equations to be equal to each other.

$s^{2} = \pi r^{2}$

We can simplify this by square rooting both sides to get s by itself:

$s = \sqrt{\pi r^{2}}$

Now we'll find the perimeter of the square and the circumference of the circle. Use the following formulas for the perimeter and circumference:

Square
$P = 4s$

Circle
$C = 2 \pi r$

The perimeter and circumference must both equal 340, so set the equations to add together to become 340:

$4s + 2 \pi r = 340$

Because we know what s equals, we can plug it into this equation:

$4( \sqrt{\pi r^{2}} ) + 2 \pi r = 340$

Simplify the equation by dividing both sides by 2 and extracting r from the first term.

$2r( \sqrt{\pi} ) + \pi r = 170$

Assuming pi = 3.1416, replace pi with this decimal value and solve:

$\sqrt{3.1416} = 1.772$

$2r(1.772) + 3.1416r = 170$

$3.544r + 3.1416r = 170$

$6.6856r = 170$

$170 \div 6.6856 = 25.4277$

$r = 25.4277$

Rounded to the nearest tenth, the radius of the circle is 25.4.

We can now use the radius to find the circumference of the circle:

$2\pi(25.42)$

$50.84(3.1416) = 159.7189$

Rounded to the nearest tenth, the circumference of the circle is 159.7 inches.

Plug this value into the original equation for the perimeter and circumference:

$4s + 159.7 = 340$

Subtract 159.7 from both sides.

$4s = 180.3$

The perimeter of the square is 180.3 inches.