Question

you have a wire that is 95 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? the circumference of the circle is cm. give your answer to two decimal places question help: d post to forum

Explanation:

Step1: Define variables

Let the circumference of the circle be \(x\) cm. Then the perimeter of the square is \((95 - x)\) cm.

Step2: Express areas in terms of \(x\)

The radius of the circle \(r=\frac{x}{2\pi}\), and the area of the circle \(A_{circle}=\pi r^{2}=\pi(\frac{x}{2\pi})^{2}=\frac{x^{2}}{4\pi}\).
The side length of the square \(s = \frac{95 - x}{4}\), and the area of the square \(A_{square}=(\frac{95 - x}{4})^{2}=\frac{(95 - x)^{2}}{16}\).
The total area \(A=\frac{x^{2}}{4\pi}+\frac{(95 - x)^{2}}{16}\).

Step3: Differentiate \(A\) with respect to \(x\)

Using the power rule \((u^{n})^\prime=nu^{n - 1}u^\prime\), \(A^\prime=\frac{2x}{4\pi}+\frac{2(95 - x)(- 1)}{16}=\frac{x}{2\pi}-\frac{95 - x}{8}\).

Step4: Set \(A^\prime = 0\) to find the critical point

\(\frac{x}{2\pi}-\frac{95 - x}{8}=0\).
Multiply through by \(8\pi\) to clear the fractions: \(4x-\pi(95 - x)=0\).
Expand: \(4x-95\pi+\pi x = 0\).
Factor out \(x\): \(x(4 + \pi)=95\pi\).
Solve for \(x\): \(x=\frac{95\pi}{4+\pi}\).

Step5: Calculate the value of \(x\)

Substitute \(\pi\approx3.14\), \(x=\frac{95\times3.14}{4 + 3.14}=\frac{298.3}{7.14}\approx41.8\).

Answer:

\(41.8\)