Problema Solution
A packaging company is going to make open top boxes with a square base that holds 100 cubic inches. What are the dimensions of the box that can be built with the least material?
Answer provided by our tutors
let 'x' be the side of the base and 'y' be the height
the volume of the box is x^2y = 100
y = 100/(x^2)
the area of the box is
A = x^2 + 4xy
A = x^2 + 4x*100/(x^2)
A = x^2 + 400/x
we need to find the minimum of the function A
dA/dx = 2x - 400x^(-2)
2x - 400x^(-2) = 0
by solving the equation
we find
x = 2*5^(2/3)
to see the step by step solution click here:
x = 5.85 cm
y = 100/(5.85^2)
y = 2.92 cm
the dimensions of the box with least material are: 5.85 cm side of the base and 2.92 cm height of the box.