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marin [14]
2 years ago
5

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 o

ther 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
Mathematics
1 answer:
Drupady [299]2 years ago
8 0

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

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