How much siding do I need and how can I minimize waste for my barn repair?

[cydetu]

Our teacher gave us this word problem:

Dear Calculus Students:

I let my barn repairs go for too long, and now I find myself in a predicament. Although the barn is structurally sound and the roof is fine, the siding is rotting. I need some help deciding on the best way to proceed, and your enterprising and resourceful professor referred me to you.

Mr. Haney has offered to sell me aluminum siding that is either three feet, two and a half feet, or one and a half feet wide. I can order the siding in any length I need, and the cost is $4.25 per square foot. Since I can get Eb from over at the Douglas place to do the labor for next to nothing, my only real expense is the siding from Mr. Haney. I want to do the entire barn in the same width siding, so for each width, I need to know how much siding I should order and how much it will cost.

I know that there will be some waste material due to the shape of the barn. Being the environmentally aware individual that I am, I'd like to know how much siding will be wasted, and I'd also be interested in any suggestions you have for what I can with with the leftovers.

I've included a sketch of the barn. Mr. Haney has only guaranteed his prices through May 2, so I would greatly appreciate your report by then.

http://www2.wheatonma.edu/academic/academicdept/MathCS/faculty/tratliff/writing/calculus-I/barn.gif

I tried to solve it but got stuck many times - any kind of help will be greatly appereciated.

TIA

 

[Gagle The Terrible]

The thing I would like to know is if you are supposed to know the equation of the 30 *45 wall. It seems like a parabola but is it ?
If you don't have the equation of the roof's curve, I really don't see how you can proceed analyticaly.

 

[HallsofIvy]

Well, setting up a coordinate system with (0,0) at the center base of the end wall, The roof is a parabola (I guess) with vertex at (0, 45), passing through (-15,30) and (15,30). That tells us that the parabola is symmetric about x=0 and so is of the form y= ax²+ 45. Setting x= 15 and y= 30, we get 30= 225a+ 45 so a= -15/225= -1/15. Once you know that you can calculate the area of that side of the barn.
The area of the sides, since they are rectangles, is easy. Since you can buy the siding at any length, there should be no waste there. Calculating the waste on the endwalls, once you are into the parabolic section, you can buy the length that reaches the end of the wall at the bottom and the waste is the part that reaches outside the parabola above that. You can reduce that by using the narrowest ( 1 1/2 feet wide) siding.

(How is this connected with "Riemann sums"?)