Math help

[Nyima]

This problem is bugging the hell out of me. I can't seem to get worked out right.

Minimize the cost of constructing a cylindrical storage tank. The tank much hold 57600 Gallons(7700 cubic feet) and be constructed from two rectangular sheets of steel. The sheet that is rolled to form the cylinder walls costs 2.75 per square foot. The sheet that forms the circular ends of the cylinder costs 5.50 per square foot. The cutting of the circles out of the rectangle costs .50 per linear foot. The rolling of the rectangular sheet into a cylinder costs .60 per linear foot. The welding of the pieces to make the cylinder costs .70 per linear foot. The tank must have two coats of rust of rust preventer on the inside of the tank and one on the outside, and it must have two coats of metallic paint on the outside of the tank. The rust preventer costs .25 per sq foot, and the metallic paint costs .40 per sq foot.


I've done problems like this before but never with so many damn variables. I know I have to take the derivative and set to zero etc etc, I just can't set up the equation.

The teacher mentioned that it should come out to a 4th power polynomial or some such.

Any help would be much appreciated.

[Silviya]

I have to go to work in a couple of minutes so I don't have time to elaborate too much, but I would start by figuring out how each cost will relate to the radius of the circle(s). Alternatively, you could use the length of the tank instead of the radius, but I think that would be slightly more work. Once you have all the costs with relation to one variable (the radius) it should be pretty simple to solve.

Use your given: tank volume must = 7700 cubic feet to find the relationship between the radius of the circle(s) and the length of the tank.

hope this helps

[Cadsuane]

This sort of question is more time consuming than challenging, and very easy to make errors in computation. First step is to clear away the rubble and organize stuff. Here we go:

The tank much hold 57600 Gallons(7700 cubic feet)

V = πr^2h = 7700

The sheet that is rolled to form the cylinder walls costs 2.75 per square foot.

2.75*2πrh

The sheet that forms the circular ends of the cylinder costs 5.50 per square foot.

2*(5.50)r^2

The cutting of the circles out of the rectangle costs .50 per linear foot.

2*(0.5)2πr

The rolling of the rectangular sheet into a cylinder costs .60 per linear foot.

2πr(0.60)

The welding of the pieces to make the cylinder costs .70 per linear foot.

2*2πr(0.70)

The tank must have two coats of rust of rust preventer on the inside of the tank and one on the outside... The rust preventer costs .25 per sq foot

3*(2πr^2(0.25)+2πrh(0.25))

and it must have two coats of metallic paint on the outside of the tank... the metallic paint costs .40 per sq foot.

2*(2πr^2(0.40)+2πrh(0.40))

Now we take the sum of these cost functions and collect terms:

C = 8.6πrh + 11.5πr + (11+2.1π)r^2
(I might've made some mistake collecting terms, but it's your homework not mine)

Recall that πr^2h=7700, and solve for either r or h (h is easier):

h = 7700/(πr^2)

Now you substitute all the h's in the cost function for 7700/(πr^2), and minimize that function. I'm sure you can take it from here. You'll get a rational function.