A box with a square base and open top must have a volume of 32000 cm3....

A box with a square base and open top must have a volume of 32000 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

Find the following:
1. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible.

2. Next, find the derivative, A'(x).

3. Now, calculate when the derivative equals zero, that is, when A'(x)=0.

4. We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x).

5. Evaluate A"(x) at the x-value you gave above.

Expert Solution

milcah answered 2 years ago

A box with a square base and open top must have a volume of 32000 cm^3....

A box with a square base and open top must have a volume of 32000 cm^3. Find the dimension of the box that minimize the amount of material used. (show all work)

A rectangular box with a square base and an open top and a volume of 1ft^3...

A rectangular box with a square base and an open top and a volume of 1ft^3 is to be made. Suppose the material used to build the sides cost $4 per ft^2 and the material used to build the bottom costs $1 per ft^2. Determine the dimensions (i.e. the side-length of the base and the height) of the box that will minimize the cost to build the box. Note: if we let x denote the side-length of the base and...

A company plans to design an open top rectangular box with square base having volume 4...

A company plans to design an open top rectangular box with square base having volume 4 cubic inches. Find the dimension of the box so that the amount of materiel required for construction is minimal. (a) Find the dimension of the box so that the amount of materiel required for construction is minimized. (b) What is the minimized material required for the construction?

Suppose a tin box is to be constructed with a square base, an open top and...

Suppose a tin box is to be constructed with a square base, an open top and a volume of 32 cubic inches. The cost of the tin to construct the box is $0.15 per square inch for the sides and $0.30 per square inch for the base. The minimized cost of the tin box is: A. $3.50 B. $$4.82 C. none of the answers D. $9.07 E. $$\$0$$

A rectangular tank with a square base, an open top, and a volume of 1372 ft...

A rectangular tank with a square base, an open top, and a volume of 1372 ft cubed is to be constructed of sheet steel. Find the dimensions of the tank that has the minimum surface area.

Minimizing Packaging Costs If an open box has a square base and a volume of 91...

Minimizing Packaging Costs If an open box has a square base and a volume of 91 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.) - Height - Length -Width

A rectangular box is to have a square base and a volume of 40 ft3. If...

A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.35 per square foot, the material for the sides costs $0.05 per square foot, and the material for the top costs $0.15 per square foot, determine the dimensions of the box that can be constructed at minimum cost. = Length Width Height how do i find length width and height

a rectangular open tank is to have a square base, and its volume is to be...

a rectangular open tank is to have a square base, and its volume is to be 125 cubic metrrs. the cost per square meter for the bottom is $24 and for the sides is $12. find the dimensions of the tank for which the cost of the material is to be the least. what is the least cost?

An open-top rectangular box is being constructed to hold a volume of 400 in3. The base...

An open-top rectangular box is being constructed to hold a volume of 400 in3. The base of the box is made from a material costing 7 cents/in2. The front of the box must be decorated, and will cost 10 cents/in2. The remainder of the sides will cost 2 cents/in2. Find the dimensions that will minimize the cost of constructing this box. Front width: Depth: Height:

An open-top cylindrical container is to have a volume of 729 cm3. What dimensions(radius and height)...

An open-top cylindrical container is to have a volume of 729 cm3. What dimensions(radius and height) will minimize the surface area?

Question