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)r²
B)(pi)r² + 10
C)(pi)r² + 1/4([pi]² * r²)
D)(pi)r² + (40 - 2[pi] * r)²
E)(pi)r² + (10 - 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)r²
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)r² + [
10 - (pi)r/2]², which is the same as
E
Cheers,
Brent
