Question

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.)

Answer 1

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

$\pi =3.14$

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

$x=3.36\ units$