Q:

# 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?

Accepted Solution

A:
Answer:16.71 cmStep-by-step explanation:GivenLength of wire L=38 cmOne piece is bent in the form of square and another in the form of circlelet x be the length of circle therefore length of square side $$\frac{38-x}{4}$$A=total area of square and circleradius 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$$x=\frac{38\pi }{4+\pi }=\frac{119.396}{7.142}=16.717 cm$$