A storage box with a square base must have a volume of 90 cubic centimeters.

Question

The top and bottom cost $0.60 per square centimeter and the sides cost $0.30 per square centimeter. Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)

Answer 1

V= 90 cm^3 = Lwh = length times width times height of the box

90 = hx^2 for a square base with sides x by x = L by w, height h= y

Cost = C = 2(.6x^2) + 4(.3xy)

y=90/x^2

C = 1.2x^2 + 1.2x(90/x^2)

= 1.2x^2 +108/x^2

take the deriviative of C with respect to x and set = 0

C'(x) = 2.4x - 108/x^3 = 0

multiply by x^3

2.4x^4 - 108 = 0

2.4x^4 = 108

x^4 = 108/2.4 = 18/.4 = 180/4 = 45

x^4 = 4th root of 45 = about 2.59 cm for sides of the base

x^2 = square root of 45 = sqr45 = 3sqr5

h = 90/x^2 = 90/sqr45 = 90/3sqr5 = 30/sqr5 = 6sqr5= about 13.42 cm for height of the box

dimensions that minimize cost are about

2.59 by 2.59 by 13.42 centimeters

C = 1.2x^2 + 1.2(90/x^2)

= 1.2(3sqr5) +1.2(90/3sqr5)

= 3.6sqr5 + 36/sqr5

=

= about $88.55 = minimum Cost for the box

find the cost of each side, sum them

put them in terms of one variable

take the derivative of the cost with respect to that one variable

set it = 0

solve for that one variable

then use it to solve for the remaining variables

(no guarantees the above calculations are error free, but it has the basic general method to find the solution. If someone else calculates the same answer or close to it, it's probably correct)