A building is the shape of an airplane hanger (a half cylinder). Construction for the foundation (rectangular part) costs $30 per square foot, the sides (semicircle bases) cost $20 per square foot and the roofing cost $15 per square foot. The volume of the entire building is 225,000 cubic feet. What should the dimensions of the building be to minimize the cost?

For my answer I got:
radius: 49.6117
height (in this case length): 58.1962

Could someone confirm my answers? Thank you very much.

Assistance needed.

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How did you get those answers?


For a fixed volume, radius is a function of length.

(1/2) pi r^2 L = 225,000
L (r) = 450,000/(pi r^2)

Cost = L*2r*30 + pi*r^2*20 + pi*r*L*15
Write cost as a function of one variable and set the derivative equal to zero. Then solve for the unknown dimension.

If that is what you did, and no mistakes were made, than your answers are correct

To confirm your answers, we need to calculate the dimensions that minimize the cost of the building.

Let's break down the problem into steps:

Step 1: Calculate the volume of the semicircular bases.
The volume of a cylinder is given by the formula: V = π * r^2 * h, where r is the radius and h is the height.
Since the building is a half cylinder, the volume of each semicircular base is half of the volume of a full cylinder. Therefore, each semicircular base has a volume of: V_base = (π * r^2 * h) / 2.

Step 2: Calculate the surface area of the rectangular part.
The rectangular part is the foundation of the building, and its surface area can be calculated as: A_foundation = length * width.

Step 3: Calculate the surface area of the sides.
The sides of the building are the curved surfaces of the semicircular bases. The surface area of each semicircular base can be calculated as: A_side = 2 * π * r * h.

Step 4: Calculate the surface area of the roofing.
The roofing is the top part of the building which is in the shape of a half cylinder. The surface area of the roofing can be calculated as: A_roofing = π * r^2.

Step 5: Write an equation for the total cost.
The total cost can be calculated as: Cost = cost_foundation * A_foundation + cost_side * A_side + cost_roofing * A_roofing.

Step 6: Substitute the given volume and costs, and eliminate the variables.
Using the formula for volume, we can substitute the expressions for A_side and A_roofing in the equation for the total cost. This will eliminate the variables A_side, A_roofing, and A_foundation.

Step 7: Find the minimum cost.
Differentiate the equation for Cost with respect to one of the dimensions and set it equal to zero. Then solve for that dimension.

Following these steps, we need more information about the cost of the foundation, sides, and roofing to find the dimensions that minimize the cost.

To confirm your answers, let's break down the problem step by step and calculate the cost of the building for the given dimensions.

Let's assume the radius of the semicircular bases is represented by "r," and the height or length of the rectangular part is represented by "h."

1. Volume of the cylinder (building):
The volume of a cylinder is given by the formula V = πr²h. In this case, V = 225,000 cubic feet. So, we have πr²h = 225,000.

2. Cost of the foundation (rectangular part):
The area of the foundation is given by A_foundation = length × width. Since the width is the same as the diameter of the semicircle (2r), the length is (h - 2r). Therefore, A_foundation = (h - 2r) × (2r).

3. Cost of the sides (semicircle bases):
The area of each semicircle base is given by A_side = 0.5 × πr².

4. Cost of the roofing:
The area of the roofing is the same as the area of the semicircles, as the roofing covers the top of the building. So, the cost of the roofing is equal to the cost of the sides.

5. Total cost:
The total cost is given by Cost = Cost_foundation + 2 × Cost_sides + Cost_roofing.

Now, let's substitute the appropriate formulas into the total cost equation and rewrite it.

Cost = (h - 2r) × (2r) × $30 + 2 × (0.5 × πr²) × $20 + (0.5 × πr²) × $15
= 60rh - 120r² + 10πr² + 7.5πr²

We have an equation for the cost in terms of "r" and "h". To minimize the cost, we can find the values of "r" and "h" that minimize the equation.

Now, you mentioned that the dimensions of the building should minimize the cost. If you calculated the dimensions to be r = 49.6117 and h = 58.1962, we can plug these values into the cost equation and calculate the minimum cost.

Cost = 60 × 49.6117 × 58.1962 - 120 × 49.6117² + 10π × 49.6117² + 7.5π × 49.6117²

By simplifying this expression, you can find the minimum cost. I recommend using a calculator or a computer program to do these calculations.