Question

Max-Min Problem help please
18. A cylinder storage tank is to contain V=16,000pi cubic feet (about 400,00 gallons). the cost of the tank is proportional to its area, so the minimal-cost tank will be the one with minimum area. The volume (V) of a cylinder of radius r and height h is (pi)(r^2)(h). its surface area is the area of top and bottom, 2(pi)(r^2), plus the side area 2(pi)(r^2)(h). find the dimensions, r and h, of the minimal-area tank.

Answer 1

The side area of the cylinder should be 2(pi)rh and not 2(pi)(r^2)(h)

Answer 2

yes sorry about that

Answer 3

V = (pi)(r^2)(h) = 16,000 ft^3
h = 16000 / ((pi)r^2) --- (1)
A = 2(pi)r^2 + 2(pi)rh = 2(pi)(r^2 + rh) ---- (2)

Put h = 6000 / ((pi)r^2) from (1) into (2).
You will have A as a function of just r (no h).
Find dA/dr, equate to 0 and solve for r.
Then put r in (1) and solve for h.

Answer 4

It should be: Put h = 16000 / ((pi)r^2) from (1) into (2).

Answer 5

Can you take it from here?

Answer 6

Yes, A = 2(pi)r^2+32000/r
Are you in calculus or pre-calculus?

Answer 7

DO you know how to find the derivative?
If not, are you allowed to use graphing calculator?

Answer 8

I am not sure what they expect you to use to minimize area A in the function:
A = 2(pi)r^2+32000/r

Answer 9

With calculus or graphing calculator, it is straightforward.

Answer 10

well graphing calculator will only be allowed 15 mins of the test

Answer 11

If you use graphing calculator, then simply plot the function:
y = 2(pi)x^2+32000/x and find the lowest point on the curve. (we are replacing r with x so it will be acceptable to some calculators).
The x value of the lowest point on the curve will be the radius that will minimize the area A. I plotted it and I got r = 13.66 feet.

Put this r value in (1) and find h.

Answer 12

27.29

Answer 13

Yes, radius = 13.66 ft and height = 27.29 ft. will have the minimum surface area.

When plotting the curve we need to choose the proper scale factor. I chose to plot x between 0 and 40 and y between 0 and 32000. And I got a good graph where the graphing calculator put a dot on the minimum point on the curve from which I was able to find x (or radius) = 13.66 ft.

Answer 14

I am answering one more question so I will show up when I am done there.

Answer 15

ok no problem

Answer 16

V = (pi)(r^2)(h) = 16,000 ft^3
h = 16000 / ((pi)r^2) --- (1)
Now there is no top.
So the area A = (pi)r^2 + 2(pi)rh
Cost C = 10(pi)r^2 + 8.64 * 2(pi)rh
C = 10(pi)r^2 + 17.28(pi)rh ---- (2)
Put h from (1) into (2)
We need to minimize C. Plot C as a function r. (or make C as y and r as x).
Find for which value of x (or r) the cost curve attains a minimum.
Then put the value of r in (1) and find h.

Answer 17

10(pi)(r^2)+276480/r

Answer 18

C = 31.415r^2 + 276480/r

Can you plot this on your graphing calculator and tell me what value of x gives the lowest point on the curve?

Answer 19

I get radius r = 16.39 feet when C will be the lowest. The big number you got is the y-coordinate or the C value and they are not asking for that.
So put r = 16.39 in (1) and find h.

Answer 20

Yes. r = 16.39 feet and h = 18.96 feet will minimize the cost.

Answer 21

thank u so much

Answer 22

you are welcome.