minimum amount of material is used in its construction.

Question

If an open box has a square base and a volume of 120 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)

Answer 1

To find the minimum amount of material used, you need the formula for SA.

 

For this open box it is the area of the square base (l * w = x * x = x²) + the area of the four sides (w * h = x * h, four times).

 

SA = x² + 4xh

 

You need to get SA into one variable, x.  So use the fact that the volume is given and the formula for the Volume of this box is l*w*h = x²h.

 

Vol = x²h = 120 in cubed.

 

So isolate "h" and then substitute it into the SA formula.   

 

h=120/(x²)

 

and SA = x² + 4 * x * [120/(x²)]

 

This simplifies to     SA = x² + 480/x

 

Now take the derivative and set it = 0 to find the x that uses the minimum amount of material.

 

d(SA)/dx = 2x - 480/(x²).     Setting this to 0, multiply both WHOLE SIDES by "x²",  and you get 2x³ - 480 = 0

 

Solving this, I get x³=240, and taking the cube root of both sides, get x = 6.21 inches.  

 

The dimensions then would be  6.21 in by 6.21 in by 3.11    (h = 120/x² = 120/6.21²)