Question

A storage company must design a large rectangular container with a square base. The volume is 2673ft

3
. The material for the top costs $8 per A square foot, the material for the sides costs $3 per square foot, and the material for the bottom costs $14 per square foot. Find the dimensions of the container that will minimize the total cost of material.

Answer 1

To minimize the total cost of material for the rectangular container, we need to find the dimensions that will minimize the surface area while satisfying the given volume requirement.

Let's assume the length of the square base is 'x' feet, and the height of the container is 'h' feet.

The volume of a rectangular container with a square base can be calculated as follows:

Volume = length * width * height

Since the base is square, the length and width are the same, so we have:

Volume = x * x * h = x^2 * h

Given that the volume is 2673 ft^3, we can write the equation:

x^2 * h = 2673

To minimize the total cost of material, we need to minimize the surface area of the container, which is composed of the top, bottom, and four sides.

The surface area can be calculated as:

Surface Area = 2(x * x) + 4(x * h)

Now we can express the surface area in terms of a single variable, x:

Surface Area = 2x^2 + 4xh

To minimize the total cost, we need to minimize the surface area. However, the given problem does not provide the cost function explicitly. Therefore, we are unable to directly optimize the total cost with respect to the surface area.

To solve this problem further and find the dimensions that minimize the total cost, we need additional information about the cost function, such as the relationship between the cost per square foot and the surface area or the specific cost associated with each component (top, bottom, sides).

Please provide more details about the cost function or any additional information for us to proceed with finding the dimensions that minimize the total cost of material.

Answer 2

To find the dimensions of the container that will minimize the total cost of material, we need to determine the dimensions of the square base and the height of the container.

Let's assume that the side length of the square base is "s" and the height of the container is "h".
Therefore, the volume of the container can be expressed as:
Volume = s^2 * h = 2673 ft^3.

Now, let's express the cost of material in terms of s and h.
The cost of the top material is $8 per square foot, so the cost of the top can be expressed as:
Cost of top = 8 * s^2.

The cost of the sides material is $3 per square foot, so the cost of the four sides can be expressed as:
Cost of sides = 4 * 3 * s * h = 12sh.

The cost of the bottom material is $14 per square foot, so the cost of the bottom can be expressed as:
Cost of bottom = 14 * s^2.

The total cost of material can be obtained by adding the cost of the top, sides, and bottom:
Total cost = Cost of top + Cost of sides + Cost of bottom.

Now we have the volume and the total cost in terms of s and h. We can use these equations to minimize the total cost.

Since the volume is given as 2673 ft^3, we can substitute s^2 * h with 2673:
2673 = s^2 * h.

Now, let's substitute the expressions for the costs into the equation for the total cost:
Total cost = 8 * s^2 + 12sh + 14 * s^2.

To minimize the total cost, we can differentiate it with respect to s and h, and set the derivatives equal to zero.

Differentiating with respect to s:
d(Total cost)/ds = 16s + 12h = 0.

Differentiating with respect to h:
d(Total cost)/dh = 12s = 0.

Solving these two equations simultaneously will give us the values of s and h that minimize the total cost.