Thread: Optimization Problem

Detergent will be packaged into a square-bottomed box with volume of 450 cubic inches. What dimensions should the box have to minimize the surface area?

I am a bit lost with this. Can someone please show me the steps on how to correctly solve this?

Thank you for any help.

Originally Posted by ilsj6

Detergent will be packaged into a square-bottomed box with volume of 450 cubic inches. What dimensions should the box have to minimize the surface area?

I am a bit lost with this. Can someone please show me the steps on how to correctly solve this?

Thank you for any help.

let x = side length of square bottom

h = box height

V = (x^2)h = 450

A = 2x^2 + 4xh

using the volume equation, solve for h in terms of x, then substitute for h in the surface area formula to get it in terms of a single variable.

find dA/dx and minimize

I understand what you wrote and that is how I solved it, initially. However, my answer just didn't seem right because that is what you would get if you solved it without Calculus. My work:

V = L^2(h)
450 = L^2(h)
h = 450/(L^2)

SA = (2L^2) + (1800L/(L^2))

I took the derivative:

SA' = 4L - (1800/(L^2))

Set it equal to zero and solved for "L:"

0 = 4L - (1800/(L^2))

-4L = - (1800/(L^2))

-4L^3 = -1800

L^3 = 450

L = 7.66

From here, I plugged it back into the volume formula to solve for "h:"

450 = 58.6756h

h = 7.669 = 7.67

Basically, my height, width, and length would approximately be 7.66 inches to minimize the Surface Area.

Is this correct?

Originally Posted by ilsj6

I understand what you wrote and that is how I solved it, initially. However, my answer just didn't seem right because that is what you would get if you solved it without Calculus. My work:

V = L^2(h)
450 = L^2(h)
h = 450/(L^2)

SA = (2L^2) + (1800L/(L^2))

I took the derivative:

SA' = 4L - (1800/(L^2))

Set it equal to zero and solved for "L:"

0 = 4L - (1800/(L^2))

-4L = - (1800/(L^2))

-4L^3 = -1800

L^3 = 450

L = 7.66

From here, I plugged it back into the volume formula to solve for "h:"

450 = 58.6756h

h = 7.669 = 7.67

Basically, my height, width, and length would approximately be 7.66 inches to minimize the Surface Area.

Is this correct?

remember your work on this ... the minimum surface area for a rectangular prism of fixed volume is a cube.

Originally Posted by skeeter

remember your work on this ... the minimum surface area for a rectangular prism of fixed volume is a cube.

Cool, so my answer is correct. I figured this would be correct...it just seemed way too easy.