Problem Solving

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) (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 + [1O - (1/2)(pi)r]^2

Here's an algebraic 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

Atekihcan wrote:Least time consuming approach is as follows...

Assume r = 0, i.e. all the wire is used to make a square.
So, perimeter of the square = 40
So, area of the square = (40/4)² = 10² = 100

So, the correct option must be equal to 100 with r = 0.

One look at the option tells us that only option E works.

Answer : E

Very nice solution, Atekihcan!

Cheers,
Brent