Pre Calculus question on homework, really struggling

Question

A hangar is formed of a half cylinder with a rectangular base and two open, semicircular ends. The curved roof costs $7 per ft2 to build. Find an expression that represents the cost per volume of the hangar (dollars/ft3) to build the curved roof. Your cost should be a function of the radius r of the semicircular end. (Round your answer to the nearest decimal place.)

 

cost per cubic foot = $??? ($/ft^3)

 

I've tried everything 

Answer 1

Here is what I can offer.  Let's write the formulas used for calculating lateral area (A_L) and volume (V) of a cylinder neglecting the areas of the bases for both...

 

A_L=2πrh.......surface area of full curved surface

                    only (open ends)

Now the hangar is a half-cylinder, so we will take half the lateral area formula...

½(A_L)=πrh...surface area of 1/2 curved surface

 

 

V=πr²h........volume of full cylinder

Now the hangar is a half-cylinder, so we will take half the volume formula...

½(V)=(πr²h)/2....half volume of cylinder

 

 

Total cost for the roof equals half lateral area times cost per square foot ($7/ft²)...

$C=½(A_L)x7....1/2 lateral area times cost

    =(πrh)x7.....substitute formula for lateral area 

    =7πrh.........total cost (dollars) in terms of "r"

                       and "h"

 

 

The question asks for cost per volume.  We can write an expression for cost over volume (a fraction) and simplify as follows...

 

  $C     7πrh       7πrh        2      14πrh    14

------ = --------- = ------  x  ------- = ------- = ----

½(V)  (πr²h)/2)      1        πr²h     πr²h      r

 

The cost will be "14/r".  The terms "π, h" and "r" in the numerator will cancel.  The unit will be "dollars per cubic foot."  I do not know why rounding is involved unless a non-multiple value of 14 for the radius is given.  Perhaps you can clarify.

 

 

Answer 2

Since the hanger is open and a half-cylinder the area = πrL where r is the radius of the open end(s) and L is the length.

 

Total cost = 7(πrL).

 

Volume = πr²L/2 since it is half a cylinder.

 

Cost / ft³ = [7πrL] / [πr²L/2]

 

Reduce this fraction and your answer will be in terms of r. You then need a value of r to get a final numeric answer.

 

 

Any questions? Comment back.