Question

The volume of a box is 120 cubic feet, the width and length are equal to x, the height is y. Find the dimensions that use the least amount of material. V=x^2 * y

Answer 1

The answer is below

Explanation:

The volume of box = width * length * height

120 = x * x * y

120 = x²y

y=120/x²

The surface area (SA) = sum of the area of all faces side

SA = x² + x² + xy + xy + xy + xy

SA = 2x² + 4xy

substitute y =120/x²

SA = 2x² + 4x(120/x²)

SA = 2x² + 480/x

Taking the derivative:

SA' = 4x - 480/x²

making SA' = 0:

0 = 4x - 480/x²

480/x² = 4x

4x³ = 480

x³ = 120

x = 4.93 feet

y = 120 / x² = 120 / 4.93² = 4.93

Hence the width, length and height is 4.93 feet.