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A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r ?

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The area of the circle is given by:

[tex]A_{C}[/tex] = πr²

[tex]A_{C}[/tex] = area, r = radius

One piece of the wire is used to form the circumference of the circle. The circumference of the circle is given by:

C = 2πr

C = circumference, r = radius

The other piece of the wire is used to form the perimeter of the square. To find the length of this piece, simply subtract C from the total length of wire, 40m:

40 - 2πr

The perimeter of the square is 4 times the length of one of its sides:

P = 4s

P = perimeter, s = side length

We know the perimeter is the length of the second piece of wire 40 - 2πr:

40 - 2πr = 4s

s = 10-πr/2

The area of the square is given by:

[tex]A_{S}[/tex] = s², where s = side length

s = 10-πr/2, so plug in s and solve for [tex]A_{S}[/tex]:

[tex]A_{S}[/tex] = (10-πr/2)²

Now that we have the areas of the circle and the square, let's add them up to find the total area:

[tex]A_{C}[/tex] + [tex]A_{S}[/tex]

= πr² + (10-πr/2)²

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