Question

You have a wire that is 38 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?
Give your answer to two decimal places

Answer 1

Answer:16.71 cm

Step-by-step explanation:

Given

Length of wire L=38 cm

One piece is bent in the form of square and another in the form of circle

let x be the length of circle

therefore length of square side $\frac{38-x}{4}$

A=total area of square and circle

radius of circle $r=\frac{x}{2\pi }$

area of circle $A_c=\pi r^2=\pi \times (\frac{x}{2\pi })^2$

Area of square $A_s=(\frac{38-x}{4})^2$

$A=\pi \times (\frac{x}{2\pi })^2+(\frac{38-x}{4})^2$

To get the minimum value of A we get

$\frac{\mathrm{d} A}{\mathrm{d} x}=\frac{2x}{4\pi }-\frac{2(38-x)}{16}$

$\frac{\mathrm{d} A}{\mathrm{d} x}=0$

$\frac{x}{4\pi }=\frac{38-x}{16}$

$x=\frac{38\pi }{4+\pi }$

Therefore circumference of circle

$x=\frac{38\pi }{4+\pi }=\frac{119.396}{7.142}=16.717 cm$