A juice factory needs to build a tank capable of holding 800pm^3 of juice. The tank will be cylindrical with the circular sides on the top and bottom. The met

Question

A juice factory needs to build a tank capable of holding 800πm³ of juice.  The tank will be cylindrical with the circular sides on the top and bottom.  The metal needed to make the curved surface of the tank will cost $32/m², while the top and bottom will be made of lower quality metal which costs $25/m².  We wish to find the value of r and h which minimize the cost of the tank.

 

For this question you may need the following formulas.  The volume of a cylinder is V=πr²h.  The area of the top or bottom of a cylinder(area of a circle) is A=πr²,  and the area of the curved surface of a cylinder is A=2πrh.

 

a) Find the formula for the total cost of the tank in terms of the radius r and the height h.

 

b) Use the given value for the volume of the tank to find a formula for h in terms of r.

 

c) Plug the answer for part (b) into the answer for part (a) to get a formula for the total cost in terms of the variable r.

 

d) Use a graphing calculator to find the value of r which minimizes the total cost. (Hint: use the window Xmin=-10, Xmax=10, Ymin=0, Ymax=2000)

 

e) Use the answer to part (d) to get the value of h which minimizes the total cost.

 

 

 

Answer 1

Covert to same units:

800pm³ = 800 * 10^-12m³

 

a) Total cost = ($32/m²)(2πrh) + ($25/m²)(2πr²)

b) V = πr²h = 800 * 10^-12m³ ⇒ h = (800 * 10^-12m³)/(πr²)

c) Total cost = ($32/m²)(2πr((800 * 10^-12m³)/(πr²))) + ($25/m²)(2πr²)

d) You can do this.

e) Once you solve for r you can plug it back into b) to solve for h.