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?

A. πr²
B. πr² +10
C. πr² + 1/4 π²r²
D. πr² + (40-2πr)²
E. πr² + (10 - (1/2)πr)²

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We are given that 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.

Since the circumference of a circle with radius R is 2Ï€R, the length of wire used to form the circle is 2Ï€R. Thus, we have (40 - 2Ï€R) left over to form the square. In other words, the perimeter of the square is (40 - 2Ï€R). However, since we need to calculate the total area of the circular and the square regions, we need to determine the side of the square in terms of R. Since the perimeter of the square is (40 - 2Ï€R), the side of the square is:

side = (40 - 2Ï€R)/4

side = 10 - (1/2)Ï€R

Now we can determine the areas of the circle and the square.

Area of circle = πR2

Area of square = side^2 = (10 - (1/2)Ï€R)^2

Thus, the combined area of the circle and square is πR2 + (10 - (1/2)πR)2.

Answer: E