Mathematics, 23.11.2019 05:31 rcmhargoux9546

Arectangular box is to have a square base and a volume of 20 ft3. if the material for the base costs $0.35 per square foot, the material for the sides costs $0.10 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.
a. length ft
b. width ft
c. height ft.

Answers: 2

Show answers

Answers

Answer from: ashley968

The volume of a box is the amount of space in it.

The dimensions that minimize cost are:

$\mathbf{Length = 2ft}$

$\mathbf{Width = 2ft}$

$\mathbf{Height= 5ft}$

The volume of the box is:

$\mathbf{V = 20}$

Assume the dimension of the base is x, and the height is h.

The volume of the box will be:

$\mathbf{V = x^2h}$

So, we have:

$\mathbf{x^2h = 20}$

The area of the sides is:

$\mathbf{A_1 = 4xh}$

The cost of the side material is 0.10.

So, the cost is:

$\mathbf{C_1 = 0.10 \times 4xh}$

$\mathbf{C_1 = 0.4xh}$

The area of the base is:

$\mathbf{A_2 = x^2}$

The cost of the base material is 0.35.

So, the cost is:

$\mathbf{C_2 = 0.35x^2}$

The area of the top is:

$\mathbf{A_3 = x^2}$

The cost of the top material is 0.15.

So, the cost is:

$\mathbf{C_3 = 0.15x^2}$

So, the total cost is:

$\mathbf{C = 0.4xh + 0.35x^2 + 0.15x^2}$

$\mathbf{C = 0.4xh + 0.5x^2}$

Recall that: $\mathbf{x^2h = 20}$

Make h the subject

$\mathbf{h = \frac{20}{x^2}}$

Substitute $\mathbf{h = \frac{20}{x^2}}$ in $\mathbf{C = 0.4xh + 0.5x^2}$

$\mathbf{C = 0.4x \times \frac{20}{x^2} + 0.5x^2}$

$\mathbf{C = \frac{8}{x} + 0.5x^2}$

Differentiate

$\mathbf{C' = -\frac{8}{x^2} + x}$

Set to 0

$\mathbf{-\frac{8}{x^2} + x = 0}$

Rewrite as:

$\mathbf{x = \frac{8}{x^2}}$

Multiply both sides by x^2

$\mathbf{x^3 = 8}$

Take cube roots of both sides

$\mathbf{x = 2}$

Recall that:

$\mathbf{h = \frac{20}{x^2}}$

So, we have:

$\mathbf{h = \frac{20}{2^2}}$

$\mathbf{h = 5}$

Hence, the dimensions that minimize cost are:

$\mathbf{Length = 2ft}$

$\mathbf{Width = 2ft}$

$\mathbf{Height= 5ft}$

Another question on Mathematics

Mathematics, 21.06.2019 16:50

If m 17 27 90 63 ** picture is attached

Answers: 1

Answer

Mathematics, 21.06.2019 18:40

The table shows the results of three plays in a football game. what is the net result of the three plays? football game 1st play 5 yards 2nd play -9 yards 3rd play 12 yards

Answers: 2

Answer

Mathematics, 21.06.2019 19:20

Part a: sam rented a boat at $225 for 2 days. if he rents the same boat for 5 days, he has to pay a total rent of $480. write an equation in the standard form to represent the total rent (y) that sam has to pay for renting the boat for x days. (4 points)

Answers: 1

Answer

Mathematics, 22.06.2019 01:10

Awavelength of 610 nm was found to be the most intense wavelength emitted by copper. answer the following questions about this wavelength of light. a) what is this wavelength in cm? b) what is the frequency that corresponds to this wavelength? c) what is the energy of light of this wavelength? d) what is the wavenumber corresponding to this wavelength?

Answers: 2

Answer

You know the right answer?

Arectangular box is to have a square base and a volume of 20 ft3. if the material for the base costs...

Questions

Mathematics, 17.02.2021 20:40

Si se daba que el volumen de un cubo es de 8cm cúbicos ¿cuál es el área del cubo?...

Mathematics, 17.02.2021 20:40

Max is designing a bedroom. The bedroom needs to be 30 cubic meters when finished. What might the dimensions be? Give two answers...