Calc. homework :)

Question

consider a wire of length 4 ft. the wire is to be cut into two pieces of length x and 4 - x. suppose the length x is used to form a circle of radius r and the length 4 - x is used to form a square with side of length a. what value of x will minimize the sum of their areas

Answer 1

area of the circle = pi(r)^2 where r= the circle's radius

pi(2r) = x= circumference of the circle

r= x/2pi

area of the circle = pi(x/2pi)^2 = (x^2)/4pi

area of the square = a^2

where 4a = 4-x

= perimeter of the square

and where a= one side of the square

a= (4-x)/4 = 1-x/4

(1-x/4)^2 =1 - x/2 + (1/16)x^2 = area of the square

total area = A= (1/4pi)x^2 + 1 - (1/2)x + (1/16)x^2

take the derivative and set equal to zero

A'(x) = x/2pi -1/2 +x/8 = 0

(1/2pi + 1/8)x = 1/2

multiply by 8pi to remove fractions

(4 +pi)x = 4pi

x = 4pi/(4+pi)

= about 1.76 feet for the circle

4-x = 4-4pi/(4+pi)

= about 2.24 feet for the square

about 1 3/4 feet for the circle and about 2 1/4 feet for the square

exactly 4pi/(4+pi) feet for the circle

and exactly 4 - 4pi/(4+pi) feet for the square

minimum total area = about 1.76 times 2.24 = about 3.94 square feet

minimum total area = exactly [4pi/(4+pi)][4-4pi/(4+pi)] square feet

Answer 2

The area of the circle (A_C) is πr² but the circle is made of a length of wire that is "x" long, in other words the circumference is "x". Since C = 2πr then x = 2πr or r = x/(2π). Therefore the area of the circle is:

A_C = π(x/(2π))²

A_C = π(x²/(4π²)

A_C = x²/(4π)

The area of the square (A_S) is a² but the square is made from a length of wire that is "4 - x" long therefore each side is (4 - x)/4 = 1 - x/4. Therefore the area of the square is:

A_S = (1 - x/4)²

A_S = (1 - x/4)(1 - x/4)

A_S = 1 - x/2 + x²/16

That means the sum of their areas is: A_C + A_S = x²/(4π) + 1 - x/2 + x²/16 or

A(x) = (1/(4π) + 1/16)x² - x/2 + 1

To minimize, take the derivative and set it equal to zero:

A'(x) = 2(1/(4π) + 1/16)x - 1/2

2(1/(4π) + 1/16)x - 1/2 = 0

2(1/(4π) + 1/16)x = 1/2

(1/(4π) + 1/16)x = 1/4

x = 0.25/((1/(4π) + 1/16)

x ≈ 1.76 ft