Optimization Problem

[Sabsssy]

I'm a little stuck on this question. Can't seem to figure out the proper equations

A fuel tank is being designed to contain 200 m3 of gasoline. The
design of the tank calls for a cylindrical part in the middle with hemispheres at
each end. If the hemispheres are twice as expensive per unit area as the
cylindrical wall, then find the radius and height of the cylindrical part so that the
cost of manufacturing the tank will minimal.

Thanks in advance!

 

[Deleted member 4993]

I'm a little stuck on this question. Can't seem to figure out the proper equations

A fuel tank is being designed to contain 200 m3 of gasoline. The design of the tank calls for a cylindrical part in the middle with hemispheres at
each end. If the hemispheres are twice as expensive per unit area as the cylindrical wall, then find the radius and height of the cylindrical part so that the cost of manufacturing the tank will minimal.
Thanks in advance!

First calculate the volume of the tank where the cylindrical part of the tank is L m long and the diameter of the hemisphere is D m.

Then calculate the surface areas of the cylindrical part and the surface area of the hemispherical part and respective costs of those parts.

Continue.....

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[Sabsssy]

First calculate the volume of the tank where the cylindrical part of the tank is L m long and the diameter of the hemisphere is D m.

Then calculate the surface areas of the cylindrical part and the surface area of the hemispherical part and respective costs of those parts.

Continue.....

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

Sorry I posted the reply twice. Still trying to get familiar with the site.

r=radius L=length
Total Volume:
200= 4/3pi r^3 + pi r^2 L
L= 200/pi r^2 - 4/3r
L'= -400/pi r^3 - 4/3
when I solve for R it gives me a negative number which is not possible as it is a length

Surface Area = 2pi r^2L + 4pi r^2

Im not sure how to incorporate the price differences and I assume I have to plug in the radius to solve for L

 

[Deleted member 4993]

Sorry I posted the reply twice. Still trying to get familiar with the site.

r=radius L=length
Total Volume:
200= 4/3pi r^3 + pi r^2 L
L= 200/pi r^2 - 4/3r
L'= -400/pi r^3 - 4/3
when I solve for R it gives me a negative number which is not possible as it is a length

Surface Area = 2pi r^2L + 4pi r^2

Im not sure how to incorporate the price differences and I assume I have to plug in the radius to solve for L

Why are you differentiating L here? You are NOT trying to minimize L - you are trying to minimize cost.

Assume that the cost of the material (per unit area) in the cylindrical part 'Q'.

Then total cost of material C = Q * 2 * pi * r^2 * L + 2* Q * 4 * pi * r^2 ........(1)

You know

200= 4/3pi r^3 + pi r^2 L

L = (200 - 4/3pi r^3) / (pi r^2) ....................................................................(2)

Now replace 'L' in (1) using (2)

Now differentiate 'C' to get minimum cost.