To minimize the amount of material used for a box with a square base and an open top while maintaining a volume of 32000 cm³, we start by defining some variables:
- Let x be the length of a side of the square base.
- Let h be the height of the box.
The volume of the box is given by the formula:
V = x² * h
Given that the volume V is 32000 cm³, we can express h in terms of x:
h = 32000 / x²
Next, we need to find the surface area A of the box, which we want to minimize. Since the box has an open top, the surface area is given by:
A = x² + 4xh
Substituting the expression we found for h into this formula gives us:
A = x² + 4x(32000 / x²)
Simplifying this, we can write:
A = x² + 128000 / x
To find the value of x that minimizes the surface area, we take the derivative of A with respect to x and set it to zero:
A’ = 2x – 128000 / x²
Setting the derivative equal to zero:
2x – 128000 / x² = 0
This simplifies to:
2x³ = 128000
So, we can solve for x:
x³ = 64000
x = 40 cm
Now that we have x, we can find h:
h = 32000 / (40²) = 20 cm
Thus, the dimensions of the box that minimize the amount of material used are:
- Base side length x = 40 cm
- Height h = 20 cm