Optimizing Cost of Half Cylinder Structure: 225K Vol

[sjnt]

Homework Statement

- Building a half cylinder structure.
- The structure must have an exact volume of 225,000 cubic feet.
- The current construction costs for the foundation are $30 per square foot, the sides cost $20 per square foot, and the roofing costs $15 per square foot.
- Minimize the cost of the structure.
- What should dimensions of the building should be to minimize the total cost?

Homework Equations

Would these be the right equations to use?
V=(pi*r^2*L)/2
SA=pi*r^2+2*r*L+pi*r*L

The Attempt at a Solution

225,000=(pi*L*r^2)/2
L=2(255,000)/pi*r^2

SA=pi*r^2+2*r*L+pi*r*L
SA=(pi*r^2)+(2*r*(2(255,000)/pi*r^2))+(pi*r*(2(255,000)/pi*r^2))
C=20$(pi*r^2)+30$(2*r*(2(255,000)/pi*r^2))+$15(pi*r*(2(255,000)/pi*r^2))

Is this right so far? I'm confused as to what to do next!

 

[dacruick]

I am confused about what this is looking like. if it is a half cylinder then there should be no siding, there should only be roof. Unless it is a rectangular prism with a semi-cylindrical piece on top.

 

[sjnt]

The sides are probably the half circles

 

[hotvette]

I think you are talking about a quonset hut

http://www.horizonhobby.com/Products/Default.aspx?ProdId=RIX6280410&utm_source=froogle

You are trying to minimize cost, so you need an equation for cost, which is the sum of the costs for the foundation area, ends, and roof. The two variables are radius and length. But, there is a relationship between radius and length in that the volume is fixed. In the end you'll have one equation for cost as a function of r that you can differentiate with respect to r, set equal to zero, and solve for r. Once you know r, you can calculate L from the volume equation.

 

[sjnt]

Ok. Would this be the formula for the sum of the costs?
SA=pi*r^2+2*r*L+pi*r*L
L=2(255,000)/pi*r^2
C=20$(pi*r^2)+30$(2*r*(2(255,000)/pi*r^2))+$15(pi*r*(2(255,000)/pi*r^2))

 

[hotvette]

Oops, yes, I didn't see that you had it before, though you should simplify the last term (pi and r in both numerator and denominator). Next step: differentiate cost equation with respect to r, set equal to zero, and solve for r.

 

[sjnt]

Ok I differentiated the equation and got:
C'(r)=40*pi*r - 27,000,000/(pi*r^2) - 6,750,000/r^2
The dimensions are:
r= 41.9385, L=81.4398

Part 2 asks:
- The cost of the flooring and siding are stable, but the roofing material has been fluctuating.
- In addition to the recommendation for the price of $15 per square foot (roofing), they need a recommendation on the dimensions of the structure if the roofing costs $R per square foot.

How would I find the dimensions?

 

[hotvette]

Same way you did it before only replace 15 with R. The answers will be expressions involving R instead of numerical values.