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
Let the ENTIRE WIRE be used to form the square.
Then:
Perimeter of the square = 40.
Side = 10.
Area = 100.
Circle area + square area = 0 + 100 = 100. This is our target.
Since the circle has no area, r=0.
Now we plug r=0 into the answers to see which yields our target of 100.
Only
E works:
πr² + (10 - (1/2)πr)² = π0² + (10 - (1/2)π0)² = 0 + 10² = 100.
The correct answer is
E.
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