Jumpskin H.

asked • 07/24/17

Optimization Problem

Hello, newbie here. I need help bad y'all. I was given an optimization problem and it is due Tomorrow. I thought I had it figured out so I was about to turn it in until another classmate said the answer was different due to adding the area of the spheres on the end using 2/3piR^3. They did not know how to solve it, but knew the answer was around 21.07 or 20.07. I need a detailed explanation of how to get the answer. I have been working on it for hours and cannot figure it out and I have 5 finals this week. Thank you in advance!!

The question is "you are asked to design a fuel tank for a fighter jet that will hold 300 liters of jet fuel. The shape of the tank is a tube with rounded ends (this is where the spheres are) a cylinder with two hemispheres attached to each end. The side (the cylinder part) cost $.1 cm^2 to make while the rounded tops (hemisphere) cost $.3 per cm^2 to make. You will find out the radius of the tube that will minimize the manufacturing cost.

Find optimal radius and show your work
explain each step
explain how solution would change with the change in the total amount of fuel
 
The steps I have so far, though I do not know if they are correct are:

h=(900-4pir^3)/3pir^2
v=4/3*pir^3 +pir^2h=300000
surface area = 4pir^2 + 2pirh
cost = 30(4pir^2) + 10(2pirh)

I then plug h into cost to get 30*(4pir^2) + 10*(2pir*((900-4pir^3)/3pir^2))
simplifying I get 120pir^2 + 20pir*((900-4pir)/3)

Then taking first derivative I get 240pir +20pi*(-4pi/3)

simplifying I get 240pir - (80pi/3)

further simplification 240pir= 80pi/3

further 240pir= 83.78

further r=.11

the answer for r is somewhere around 20-21 so I know this is wrong.

Kendra F.

The area of the sphere (4πr2) is accounted for in the SA you have.
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07/24/17

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Kendra F. answered • 07/24/17

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Your height seems to be off.
 
300,000 = hπr2 + (4/3)πr3
300,000 - (4/3)πr3 = hπr2
(300,000 - (4/3)πr3)/(πr2) = h
300,000/(πr2) - (4/3)r = h
 
If you use this you get an answer close to 20, r = 21.7 cm
 
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Jumpskin H.

Thanks Kendra and Kenneth!
 
That works when you solve for r, but when I take the derivative like I know need to be done I get a much different answer. 
 
So, what I have gotten is 4/3*π*r^3 + π*r^2h
 
solving for h I get (300/π*r^2)- 4r/3
 
Substituting this into the equation along with the prices I get 1.2/3*π*r^3 + .1π*r^2 (300/π*r^2 - 4r/3)
 
Taking the derivative with repeat to r I get 1.2π*r^2 + .2π*r(300/πr^3 -4/3)
 
Setting this equal to zero I get r=1.36
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07/24/17

Kendra F.

You want everything to be in units of cm, cm2 or cm3 so that units cancel out. 
 
Here is what I did;
 
C(r) = 0.1*(2πr)*(300,000/(πr2) - 4r/3) + 0.3(4πr2)
C(r) = 60,000/r - 0.8πr2/3 + 1.2πr2
C'(r) = -60,000/r2 - 1.6πr/3 + 2.4πr
C'(r) = -60,000/r2 -1.6πr/3 + 7.2πr/3
C'(r) = -60,000/r2 + 5.6πr/3
0 = -60,000/r2 + 5.6πr/3
60,000/r2 = 5.6πr/3
180,000 = 5.6πr3
180,000/5.6π = r3
3√(180,000/5.6π) = r
21.7 cm = r
 
 
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07/24/17

Kendra F.

The answer will vary depending on if the and when any numbers are rounded. I prefer to not round until the final solution. The volume needs to be 300,000 cm3 to keep the units correct so that they cancel out leaving $ for the cost. 
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07/24/17

Jumpskin H.

Your an absolute lifesaver!!!!! Many, many thanks!
 
The only questions I have left are which equation would I plug r back into to prove it was a minimum and how would the solution change with the change in volume?
 
I assume he wants some sort of formula for change in volume question. Would I just plug V back into the equation for this instead of 300000?
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07/24/17

Kendra F.

I don't think you'll need anything for change in volume. The problem asks for the measurements of tank that minimize the cost of manufacture but is still capable of accommodating 300 L of jet fuel. It says nothing about changing volume. Of course the volume of jet fuel changes as it gets used but not the tank that contains it.
 
You would put r into the function for cost to show it produces the minimum cost of manufacturing.
C(r) = 60,000/r + 2.8πr2/3
 
 r         C(r)
19         4216.40
19.5      4191.87
20         4172.86
20.5      4159.07
21         4150.22
21.5      4146.09
21.7      41.45.70
22         4146.43
22.5      4151.07
23         4159.80
 
You can calculate the height, h for completeness.
 
300,000 = (4/3)*π(21.7)3 +π(21.7)2h
h = 173.86 cm
 
 
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07/24/17

Jumpskin H.

You the girl! You just saved my grades!
 
You are right. The project said for bonus points I could
 
"explain how the solution would change as the total amount of fuel changes"
 
but as you stated, the volume/radius would not change with a change in fuel. I assume he was not asking for how the volume change would affect radius, but it was a trick question instead (or if not I can show him the wording and he will probably give it to me).
 
Again, thank you so much for your help. I really appreciate it!
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07/24/17

Kenneth S. answered • 07/24/17

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Two hemispheres can be mentally added to get one sphere. The formula for the volume of a sphere is (4/3)pi(r3) which you must add to the volume of the cylinder to get total tank volume. The volume must be equivalent to 300 liters. 
 
The surface area of one sphere is 4(pi)r2.
 
so the difficulty I see is that you have mixed up formula for area & formula for volume.
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