Answer:
M(x) = 452,56 in
Explanation:
The volume of the open box is 500 in³
V = Area of the base times height
V(b) = x² * h where x is the side of the square and h the heigh
Then 500 = x²*h
Total material to use is: material of the base + material of 4 sides
material of the base is x²
material of one side is x*h we have 4 sides then 4*x*h
Total material M(b)
M(b) = x² + 4*x*h
And as h = 500/ x²
M(x) = x² + 4* x* 500/x²
M(x) = x² + 2000/x
Tacking derivatives on both sides of the equation
M´(x) = 2*x - 200/x²
M´(x) = 0 2*x - 200/x² = 0
x³ - 100 = 0
x³ = 100
x = 4,64 in
And By substitution h = 500/x²
h = 500/(4,64)² h = 23,22 in
How do we know that x = 4,64 make V(x) minimum
we get the second derivative
M´´(x) = 2 + 200*2x/ x⁴ = 2 + 400/x³
M´´(x) is always positive
M´´(x) > 0 then M(x) has a minimum for x= 4,64 in
The least amount of material is:
M(x) = x² + 2000/x
M(x) = (4,64)² * 2000/4,64
M(x) = 21,53 + 431,03
M(x) = 452,56 in