A closed box with a square base is to be built,the bottom and all four sides of

Question

A closed box with a square base is to be built,the bottom and all four sides of the box are to be made of material that cost 1$ per square foot and the top is to be made of glass costing 5 dollars per square foot. what are the dimensions of the box of with the greatest volume that can be constructed for $72? what is the maximum volume? verify that you found the absolute maximum A closed box with a square base is to be built,the bottom and all four sides of the box are to be made of material that cost 1$ per square foot and the top is to be made of glass costing 5 dollars per square foot. what are the dimensions of the box of with the greatest volume that can be constructed for $72? what is the maximum volume? verify that you found the absolute maximum what are the dimensions of the box of with the greatest volume that can be constructed for $72? what is the maximum volume? verify that you found the absolute maximum

Explanation / Answer

Suppose that x is the length and width is y is the height, therefore:
The area of the bottom is x^2, cost of that is x^2
The area of each side is xy, area of four sides is 4xy, cost of that is 4xy
Area of the top is x^2, cost of that is 5x^2
Cost of the box is:
x^2 + 4xy + 5x^2 = 72
6x^2 + 4xy = 72
3x^2 + 2xy = 36
2xy = 36 - 3x^2
y = (36 - 3x^2) / (2x) ------------------1
The volume of the box is:
V = x^2y
V = x^2 * (36 - 3x^2) / (2x)
V = 18x - 1.5x^3
dV/dx = 18 - 4.5x^2
Set that equal to zero to get values-
18 - 4.5x^2 = 0
4.5x^2 = 18
x^2 = 18/4.5
x^2 = 4
x = sqrt(4)
x = 2

Now , put x=2 in 1 and get y value
y = (36 - 3*2^2) / (2*2)
y = (36 - 3*4) / 4
y = (36 - 12) / 4
y = 24/4

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.