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.)

The radius of the circle is

The length of the square is

Step-by-step explanation:

we know that

The circumference of a circle is equal to

The perimeter of the square is equal to

so

Simplify

-----> equation A

The area of a circle is equal to

The area of a square is

The total area is equal to

-----> equation B  

substitute equation A in equation B

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

Find the value of x

assume

honesty mood

step-by-step explanation:

step-by-step explanation:

step 1: solve one of the equations for one of the variables. let's solve the first equation for y:

step 2: substitute that equation into the other equation, and solve for x.

step 3: substitute x = 4 x = 4 x=4 into one of the original equations, and solve for y.