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:The radius of the circle is [tex]r=1.68\ units[/tex]The length of the square is [tex]x=3.36\ units[/tex]Step-by-step explanation:we know thatThe circumference of a circle is equal to [tex]C=2\pi r[/tex]The perimeter of the square is equal to [tex]P=4x[/tex]so[tex]24=2\pi r+4x[/tex]Simplify[tex]12=\pi r+2x[/tex][tex]x=(12-\pi r)/2[/tex] -----> equation AThe area of a circle is equal to [tex]A=\pi r^{2}[/tex]The area of a square is [tex]A=x^{2}[/tex]The total area is equal to [tex]At=\pi r^{2}+x^{2}[/tex] -----> equation B  substitute equation A in equation B [tex]At=\pi r^{2}+[(12-\pi r)/2]^{2}[/tex] This is a vertical parabola open upwardThe vertex is the minimumThe x-coordinate of the vertex is the radius of the circle that produce a minimum areaThe y-coordinate of the vertex is the minimum areaSolve by graphingThe vertex is the point (1.68, 20.164)see the attached figurethereforeThe radius of the circle is [tex]r=1.68\ units[/tex]Find the value of x[tex]x=(12-\pi r)/2[/tex]assume[tex]\pi =3.14[/tex][tex]x=(12-(3.14)*(1.68))/2[/tex][tex]x=3.36\ units[/tex]