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?

Solution:

Approach 1:

A circle with radius

r will have the circumference (the length of wire used) = 2πr

The ares of the circle is πr^2

So, to crate the square you have = 40 - 2πr

This will be the perimeter of the square. So, the length of each side of the square = (40 - 2πr)/4. = 10 - (1/2)πr

The area of the square is = (10 - (1/2)πr)^2

The answer is (E).

Approach 2:

Plug in values for the variable. Let r = 0 (It does mean that there is no circle at all.....but it is completely safe to assume this)

So, that means the complete 40 meter wire is used to create the square.

The perimeter of the square is 40.

=> The length of each side = 40/4 = 10

So, the area is 10^2 = 100

Now, check the answers.

A) π(0²) = 0

B) π(0²) + 10 = 10

C) π(0²) + 1/4(π² * 0²) = 0

D) π(0²) + (40 - 2π0)² = 1600

E) π(0²) + (10 - 1/2π(0))² = 100 PERFECT!

The answer is (E).

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