Problem Solving

amirhakimi wrote:A thin piece of 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 term of r?

A) (pi) * r^2
B) (pi) * r^2 + 10
C) (pi) * r^2 + 1/4([pi]^2 * r^2)
D) (pi) * r^2 + (40 - 2[pi] * r)^2
E) (pi) * r^2 + (10 - 1/2[pi] * r)^2

Answer is E

Here's one approach:

Since r is the radius of the circle, the area of the circle will be (pi)r^2.

If r is the radius of the circle, the length of wire used for this circle will equal its circumference which is 2(pi)r

So, the length of wire to be used for the square must equal 40 - 2(pi)r

In other words, the perimeter of the square will be 40 - 2(pi)r

Since squares have 4 equal sides, the length of each side of the square will be [40 - 2(pi)r]/4, which simplifies to be 10 - (pi)r/2

If each side of the square has length 10 - (pi)r/2, the area of the square will be [10 - (pi)r/2]^2

So, the total area will equal (pi)r^2 + [10 - (pi)r/2]^2, which is the same as E

Cheers,
Brent

amirhakimi wrote:A thin piece of 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 term of r?

A)(pi) * r^2
B)(pi) * r^2 + 10
C)(pi) * r^2 + 1/4([pi]^2 * r^2)
D)(pi) * r^2 + (40 - 2[pi] * r)^2
E)(pi) * r^2 + (10 - 1/2[pi] * r)^2

Answer is E

Another approach is to plug in a value for r and see what the output should be.

Let's say r = 0. That is, the radius of the circle = 0
This means, we use the entire 40-meter length of wire to create the square.
So, the 4 sides of this square will have length 10, which means the area = 100

So, when r = 0, the total area = 100

We'll now plug r = 0 into the 5 answer choices and see which one yields an output of 100

A) (pi) * 0^2 = 0 NOPE
B) (pi) * 0^2 + 10 = 10 NOPE
C) (pi) * 0^2 + 1/4([pi]^2 * 0^2) = 0 NOPE
D) (pi) * 0^2 + (40 - 2[pi] * 0)^2 = 1600 NOPE
E) (pi) * 0^2 + (10 - 1/2[pi] * 0)^2 = 100 PERFECT!

Answer: E

Cheers,
Brent