Greatest volume?

Question

A closed box with squared base is to be built from 2 different materials. The top of the box and all 4 sides are to be made of material costing 1 dollars/ft². and the bottom is to be constructed of another material costing 3 dollars/ft² - what are the dimensions of the box of greatest volume that can be constructed for 48 dollars.

Answer 1

If a is the length of the side of the top/bottom and h is the height of the box.

Volume = h*a^2 and Cost = $3*a^2+$1*a^2+$1*4*a*h

Since cost is $48, we can plug that in to get:

$48 = 4*a^2+4*a*h = 4*a*(a+h)... This simplifies to:

$12 = a*(a+h).

From this you can get a relationship between a and h:

h = (12-a^2)/a. If you substitute this into the equation for volume, you get an expression relating volume to the length of side a:

V = ((12-a^2)/a)*a^2 = (12-a^2)*a = 12a-a^3

Take the derivative of this and set it equal to zero, which is the point where the volume stops increasing as you increase a.

dV/da = 12 - 3*a^2 = 0 ==> a = 2, h = 4

Answer 2

for the 2 previous answers, i thought a box had all sides be equal, as in squared box. why are you giving two different answers for length and height?

Answer 3

Cost = C, x = one side of base, h = height of box, V = volume, A = area.

cost is cost of bottom, plus top, plus sides.

C = x²*3 + x²*1 + 1*h*4*x

C = 3x² + x² + 4xh

48 = 4x² + 4xh

12 = x² + xh

h = (12 - x²) / x

we need to maximize the volume.

Volume = hx²

Volume = x²(12 - x²) / x

Volume = x(12 - x²)

Volume = 12x - x³

dV/dx = 12 - 3x² = 0

3x² = 12

x = 2

h = (12 - x²) / x

h = (12 - 2²) / 2 = 4

so a box with sides 2 ft and height 4 ft.

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