F ind the minimum cost of constructing a cylindrical storage tank with all of th
ID: 2864438 • Letter: F
Question
Find the minimum cost of constructing a cylindrical storage tank with all of the following requirements. The tank must hold 57,600 gallons and be constructed from two rectangular sheets of steel. The sheet that is rolled to form the cylinder walls costs $2.80 per square foot. The sheet that forms the circular ends of the cylinder costs $5.40 per square foot. The cutting of the circles out of the rectangle cost $.50 per linear foot. The rolling of the rectangular sheet into a cylinder costs $.30 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 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 cost $.40 per square foot, and the metallic paint costs $.50 per square foot. Lastly, the total fixed cost for the inflow and outflow valves, an access panel, and their installation is $1135.84.
4Explanation / Answer
Volume V = 57600 gal Converting to feet = 57600 / 7.48
= Pi * r^2 * h
So, h = 57600 / (7.48 * Pi * r^2)
Curved surface area ACS = 2 * Pi * r * h
ACS = 2 * Pi * r * 57600 / (7.48 * Pi * r^2)
Cost for forming the ACS = 2.80 * CS
Perimeter Circular lids PL = 2 * Pi * r
Area of circular lid AL = Pi * r^2
Cost for forming the lids = 5.40 * AL * 2 for 2 lids
Cutting cost = 0.5 * PL
Rolling cost (I assumed that the thing is rolled only about the circumference) = 0.3 * PL
Welding cost (I assumed Welding is done on the joining side of the cylinder) = 0.7h = 0.7 * ( 57600 / (7.48 * Pi * r^2))
The Coating cost (I assumed that the total surface area is coated) = (0.4 + 0.5)(CSA + 2(AL)
Adding everything we have here, we get
C = 2.80(2 * Pi * r * 57600 / (7.48 * Pi * r^2)) + (5.40( Pi * r^2 )* 2) + 0.5 (2 * Pi * r) + 0.3(2 * Pi * r) + 0.7(( 57600 / (7.48 * Pi * r^2)) +
(0.4 + 0.5)((2 * Pi * r * 57600 / (7.48 * Pi * r^2)) + 2(Pi * r^2))
Now differentiate C with respect to r and equate to 0 to get the minimum which is the answer.
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