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aleksklad [387]
3 years ago
14

A piece of wire of length 7070 is​ cut, and the resulting two pieces are formed to make a circle and a square. Where should the

wire be cut to​ (a) minimize and​ (b) maximize the combined area of the circle and the​ square?
Mathematics
1 answer:
lyudmila [28]3 years ago
3 0

Answer:

a.x=39.2

b.Use whole  wire as a circle

Step-by-step explanation:

We are given that

Length of piece of wire=70  units

Let length of wire used to make a square =x units

Length of wire used in circle=70- x

Side of square=\frac{perimeter\;of\;square}{4}=\frac{x}{4}

Circumference of circle=2\pi r

70-x=2\pi r

r=\frac{70-x}{2\pi}

Combined area of circle and square,A=(\frac{x}{4})^2+\pi(\frac{70-x}{2\pi})^2

Using the formula

Area of circle=\pi r^2

Area of square=(side)^2

a.A=\frac{x^2}{16}+\frac{4900+x^2-140x}{4\pi}

Differentiate w.r.t x

\frac{dA}{dx}=\frac{x}{8}+\frac{2x-140}{4\pi}

\frac{dA}{dx}=0

\frac{x}{8}+\frac{2x-140}{4\pi}=0

\frac{\pi x+4x-280}{4\pi}=0

\pi x+4x-280=0

x(\pi+4)=280

x=\frac{280}{\pi+4}

x=39.2

Again differentiate w.r.t x

\frac{d^2A}{dx^2}=\frac{1}{8}+\frac{1}{2\pi}>0

Hence, the combined area of circle and the square is minimum at x=39.2

b.When the wire is not cut and whole wire used  as a circle . Then, combined area is maximum.

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