Question

An open box with a square base is to have a volume of 18 ft^3.

(a) Find a function that models the surface area A of the box in terms of the length of one side of the base x.

(b) Find the box dimensions that minimize the amount of material used. (Round your answers to two decimal places.)

Answer 1

Answer:

2.29 ft of side length and 1.14 height

Step-by-step explanation:

a) Volume V = x2h, where x is side of square base and h is hite.

Then surface area S = x2 + 4xh because box is open.

b) From V = x2h = 6 we have h = 6/x2.

Substitude in formula for surface area: S = x2 + 4x·6/x2, S = x2 + 24/x.

We get S as function of one variable x. To get minimum we have to find derivative S' = 2x - 24/x2 = 0, from here 2x3 - 24 = 0, x3 = 12, x = (12)1/3 ≅ 2.29 ft.

Then h = 6/(12)2/3 = (12)1/3/2 ≅ 1.14 ft.

To prove that we have minimum let get second derivative: S'' = 2 + 48/x3, S''(121/3) = 2 + 48/12 = 6 > 0.

And because by second derivative test we have minimum: Smin = (12)2/3 + 4(12)1/3(12)1/3/2 = 3(12)2/3 ≅ 15.72 ft2

Answer 2

The box dimensions that minimize the amount of material used are; side length of base = 3.30 ft and height = 1.65 ft

How to minimize area?

a) We know that a box has length, width and height and so volume of a box is;

V = L × W × H

Now, we are told the base of the box is square and if we label it x, then we have to function that models the volume as;

V = x²h

where x is side of square base and h is height.

Thus surface area function is;

S = x² + 4xh

b) We are told that the volume is 18 ft³ and so;

x²h = 18

Thus;

h = 18/x².

Put 18/x² for h in the surface area function to get: S = x² + 4x(18/x²)

S = x² + 72/x.

To get minimum, we have to find derivative of S. Thus;

S' = 2x - 72/x² = 0,

Thus;

2x³ = 72

x³ = 72/2

x³ = 36

x = 3.3 ft

Thus;

h = 18/3.3²

h = 1.65 ft

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