Optimization problems

[coldcell]

A fuel tank is being designed to contain 200 m cube of gasoline; however, the maximum length tank that can be safely transported to clients is 16 m long. 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 be minimal. Give the answer correct to the nearest centimetre

V = 2/3 pi r³ h
200 = 2/3 pi r³ h
300 = pi r³ h
h = 300/ (pi)(r³)

So this is the relationship that I find between height and radius.

This is where I'm lost. The pricing of the cynlindrical wall and the hemisphere :S If I can get the formula for this one, I think I can solve the problem.

A truck crossing the prairies at a constant speed of 110 km/h gets 8 km/ L of gas. Gas costs $0.68/L. The truck loses 0.10 km/L in fuel efficiency for each km/h increase in speed. Drivers are paid $35/h in wages and benefits. Fixed costs for running the truck are $15.50/h. If a trip of 450 km is planned, what speed will minimize operating expenses?

For this one, I tried to form a relationship between the speed, time and fuel but end up confusing myself. Am I supposed to take it 1 step at a time or what? I never studied this kind of optimization problem before.

 

[soroban]

Heloo, coldcell!

Sorry, your set-up is incorrect . . .

A fuel tank is being designed to contain 200 m³ of gasoline.
However, the maximum length tank that can be safely transported to clients is 16 m long.
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 be minimal.
Give the answer correct to the nearest centimetre

Let \(\displaystyle H\) = height of cylinder, \(\displaystyle R\) = radius of cylinder.

Volume of cylinder: \(\displaystyle \,V_c\:=\:\pi R^2H\)

Volume of two hemispheres: \(\displaystyle \,V_h \:=\:\frac{4}{3}\pi R^3\)

Total Volume: \(\displaystyle \,\pi R^2 H\,+\,\frac{4}{3}\pi R^3\:=\:200\;\;\Rightarrow\;\;H\:=\:\frac{200}{\pi R^2}\,-\,\frac{4}{3}R\;\) [1]


Surface of cylindrical wall: \(\displaystyle \,A_c\:=\:2\pi RH\)
At \(\displaystyle k\) dollars per m², the cost of the cylindrical wall is:
\(\displaystyle \;\;\;C_c\:=\:2\pi R^2H\, \times\, k\:=\:2k\pi R^2H\) dollars.

Surface of two hemispheres: \(\displaystyle \,A_h\:=\:4\pi R^2\)
At \(\displaystyle 2k\) dollars per m², the cost of the two hemispheres is:
\(\displaystyle \;\;\;C_h\:=\:4\pi R^2\,\times\,2k\:=\:8k\pi R^2\) dollars.

Total Cost: \(\displaystyle \,C\;=\;2k\pi R^2H\,+\,8k\pi R^2\) dollars.


Substitute [1]: \(\displaystyle \,C\;=\:2k\pi R^2\left(\frac{200}{\pi R^2}\,-\,\frac{4}{3}R\right)\,+\,8k\pi R^2\)

\(\displaystyle \;\;\)which simplifies to: \(\displaystyle \,C\;=\;400kR^{-1}\,+\,\frac{16}{3}k\pi R^2\)

And that is the function we must minimize . . .
.

 

[galactus]

A truck crossing the prairies at a constant speed of 110 km/h gets 8 km/ L of gas. Gas costs $0.68/L. The truck loses 0.10 km/L in fuel efficiency for each km/h increase in speed. Drivers are paid $35/h in wages and benefits. Fixed costs for running the truck are $15.50/h. If a trip of 450 km is planned, what speed will minimize operating expenses?

For this one, I tried to form a relationship between the speed, time and fuel but end up confusing myself. Am I supposed to take it 1 step at a time or what? I never studied this kind of optimization problem before.

This is kind of tricky to wrap the ol' brain around, but I will give it a shot.

If we let s be the speed over 110.

The expenses for the trip are gas, wages for driver and fixed costs.

So, I believe, we have:

\(\displaystyle \L\\(\frac{450}{8-0.1s})(0.68)+(35)\frac{450}{110+s}+(15.5)\frac{450}{110+s}\)

This is the function that needs optimized. I hope.

Check me out.