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MaRussiya [10]
3 years ago
13

The sum of the circumference of a circle and the perimeter of a square is 24. Find the dimensions of the circle and square that

produce a minimum total area. (Let x be the length of a side of the square and r be the radius of the circle.)

Mathematics
1 answer:
lyudmila [28]3 years ago
8 0

Answer:

The radius of the circle is r=1.68\ units

The length of the square is x=3.36\ units

Step-by-step explanation:

we know that

The circumference of a circle is equal to C=2\pi r

The perimeter of the square is equal to P=4x

so

24=2\pi r+4x

Simplify

12=\pi r+2x

x=(12-\pi r)/2 -----> equation A

The area of a circle is equal to A=\pi r^{2}

The area of a square is A=x^{2}

The total area is equal to

At=\pi r^{2}+x^{2} -----> equation B  

substitute equation A in equation B

At=\pi r^{2}+[(12-\pi r)/2]^{2}

This is a vertical parabola open upward

The vertex is the minimum

The x-coordinate of the vertex is the radius of the circle that produce a minimum area

The y-coordinate of the vertex is the minimum area

Solve by graphing

The vertex is the point (1.68, 20.164)

see the attached figure

therefore

The radius of the circle is

r=1.68\ units

Find the value of x

x=(12-\pi r)/2

assume

\pi =3.14

x=(12-(3.14)*(1.68))/2

x=3.36\ units

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