A rope 14 feet long is cut into two pieces. One piece is used to form a circle and the other used to form a square. Find a function...

Question

A rope 14 feet long is cut into two pieces. One piece is used to form a circle and the other used to form a square. Find a function representing the area of both square and circle as a function of the length of one side of the square.

Answer 1

OOPS!

Ryan has the right solution procedure•, but he made a mistake. Here's an edited version of his answer:

The rope used for the perimeter of the square should be 4x (the square has 4 sides). The amount of rope used for the circle is (14-4x).

The area of the square is indeed x².

(14-4x) would be the circumference of the circle. Since circumference = pi • 2r, we need to rearrange (14-4x) = pi • 2r to be in terms of r. When we do this, we can then represent area (pi • r²).

We get r = (14-4x)/(2pi). When we plug this value in for r in the area formula, the area of the circle = pi • [(14-4x)/(2pi)]².

The overall function would be x² + pi x [(14-4x)/(2pi)]² = total area of square and circle.

That is an easy-to-make oversight which I have also done before. That's awfully embarrassing, so I like to proofread my solutions and re-think my way through a solution before I submit it, so I usually (but not always) catch and correct it.

Answer 2

We can make the side of the square be represented as x.

That means that the length of rope used for the circle would be (14-x).

So...

x² would be the area of the square. The circle is a bit trickier.

14-x would be the circumference of the circle. Since circumference = pi x 2r, we need to rearrange 14-x = pi x 2r to be in terms of r. When we do this, we can then represent area (pi x r²).

So after rearranging the circumference equation, we get r = (14-x)/(2pi). When we plug this value in for r in the area formula, the area of the circle = pi x [(14-x)/(2pi)]²

The overall function would be x² + pi x [(14-x)/(2pi)]² = total area of square and circle