Question

21.    Solve the problem.A 44-in. piece of string is cut into two pieces. One piece is used to form a circle and the other to form a square. How should the string be cut so that the sum of the areas is a minimum? Round to the nearest tenth, if necessary.  

a)Circle piece = 10.6 in., square piece = 33.4 in.
b)Square piece = 0 in., circle piece = 44 in.
C)Square piece = 10.6 in., circle piece = 33.4 in.
D)Square piece = 10.9 in., circle piece = 10.4 in.

Answer 1

For now, let's have the equation for the sum of the areas to be

• A = C + S

where C is the area of the circle and S is the area of the square. Remember that we will want to minimize A when we're ready. Since one part of the string will become the circumference of the circle (call it "c") and the other part will become the perimeter of the square (call is "s"), we can have the formula:

• c + s = 22

We will find what the circle area is equal to in relation to its circumference, and likewise for the square:

c = 2πr

r = c/2π

So since C = πr²

C = π(c/2π)² when you plug in what we got for r.

We do the same for the square:

s = 4x , where x is a side of the square.

x = s/4

So since S = x²

S = (s/4)² when you plug in what we got for x.

We plug these in to our A equation:

• A = π(c/2π)² + (s/4)²

And remember that c & s are related as well, such that s = 22 - c, so we plug this in so it's all dependent on one variable:

• A = π(c/2π)² + ((44-c)/4)²

Simplify:

• A = π(c²/4π²) + (1/16)(44-c)²
• A = c²/4π + (1/16)(1936 - 88c + c²)
• A = (4/16π)c² + (π/16π)c² - (44/16)c + 484/16
• A = ((4+π)/16π)c² - (88/16)c + 1936/16

So we find that the sum of the areas becomes an upwards parabola, which is based on how long we make the string for the circumference of the circle. Here, we have a fork in the road of methods, depending on which class you're in.

1. IF YOU'RE IN CALCULUS: Just take the derivative of this and set it equal to 0, and from there you can solve for c.
2. IF YOU'RE IN ALGEBRA II: The c we want to find will be the x-value of the vertex. This means that we have to change this quadratic equation from Standard Form to Vertex Form. We do this by Completing the Square.

Answer: a)Circle piece = 10.6 in., square piece = 33.4 in.