Question

You have a wire that is 74 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum

Answer 1

Answer:

32.55 cm  

Step-by-step explanation:

         Let x = the length of wire that becomes a circle.

Then 74 - x = the length of wire that becomes a square

Circumference of circle + perimeter of square = 74

1. Expression for the side of the square

P = 4s = 74 - x

s = ¼(74 - x)

2. Expression for the radius of the circle

C = 2πr = x

r = x/(2π)

3. Expression for the total area

$\begin{array}{rcl}\text{Total area} & = &\text{area of circle+ area of square}\\A & = & \pi r^{2} + s^{2}\\& = &\pi \left(\dfrac{x}{2 \pi}\right)^{2} + \left (\dfrac{1}{4}(74 - x)\right)^{2}\\\\ & = & \dfrac{x^{2}}{4 \pi} + \dfrac{1}{16}(5476 - 148x + x^{2})\\\\ & = & 0.07958x^{2} + 342.25 - 9.25x + 0.0625x^{2}\\A & = & 0.1421x^{2} -9.25x + 342.25 \\ \end{array}$

This is the equation of a parabola.

In standard form,

ƒ(x) = 0.1421x² -9.25x + 342.24

a = 0.1421; b = -9.25; c = 342.24  

The parabola opens upwards, because a > 0. Therefore, the vertex is a minimum.

The vertex of a parabola occurs at

x = -b/(2a) = 9.25/(2 × 0.1421) = 9.25/(0.2842) = 32.55

The circumference of the circle is 32.55 cm.

The graph below shows that the area of the circle is a minimum when x = 32.55 cm