F ind the minimum cost of constructing a cylindrical storage tank with all of th

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.

Explanation / Answer

Hi :)

The tank much hold 57600 gallons (7700 cubic feet)

V=pi*r^2*h=7700

The sheet that is rolled to form the cylinder cost 2.80 per square foot

2.80*2pi*r*h

The sheet that forms the circular ends of the cylinder costs 5.40 per square foot

2*5.40*r^2

the cutting of the circles out of the rectangle costs 0.50 per linear foot

2*0.5*2*pi*r

The rolling of the retangular sheet into a cylinder costs 0.30 per linear foot

2pi*r*(0.30)

The welding of the pieces to make the cylinder costs 0.70 per linear foot

2*2pi*r(0.70)

The tank must have two coats of rust preventer on the inside of the tank and one on the outside

The rust preventer costs 0.40 per square foot

3*(2pi*r^2(0.40)+2pi*r*h(0.40))

and it must have two coats of metallic paint on the outside of the tank

The metallic paint costs 0.40 per square foot

2*(2pi*r^2(0.40)+2pi*r*h(0.40))

Total fixed costs

1135.84

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

C=9.6pi*r*h+(4pi+10.8)r^2+4.2pi*r+1135.84

Recall that pi*r^2*h=7700

and solve for either r or h (h is easier)

h=7700/(pi*r^2)

Now, you substitute all the h's in the cost function for 7700/(pi*r^2) and minimize that funtion (finding the derivative and equating to zero).

I'm sure you can take it from here.

I hope it helps :)

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.