finding optimization

Question

A box manufacturing company would like to minimize the amount of material it takes to make their boxes. The boxes have widths of 10 cm and do not have any tops. Given that the volume of the box is 110 cm³, identify the length  and height  that will minimize the surface area. Round to three decimal places.

Answer 1

If L and H represent the length and height of the box (we know W is 10), then the surface area of the box can be determined in terms of L and H as follows:

Area of bottom of the box: 10L.

Area of one side involving H and 10: 10H, but there are two of those sides, so 20H.

Area of a side involving L and H: LH, but there are two of those sides, so 2LH.

So, surface area S is represented by: S = 10L + 20H + 2LH.

Use the formula V = LWH to establish a relationship between L and H, so that one of the variables in the formula for surface area can be eliminated.

110 = 10LH

11=LH

H=11/L

Replace H in the surface area formula with 11/L. Take the derivative of S with respect to L. Determine when S' is equal to zero (critical numbers). Convince yourself that that value gives a minimum value for the surface area.

Use that value of L to determine H (H=11/L).