Optimizing Fuel Use

[sanspaba]

Hello lovely's,

 

I am having some issues with my calculus project for this term - my issues mostly involves identifying what the question is asking. The overall project is about fuel optimization, and focuses on using integration. I am pretty close to completing this project, but have some minor hiccups.

 

1. The image attached is the graph given, which is a function of m(v).

[math] m(v) = 1.9ve^{-v/40} [/math]

 

Question: At what speed does the car get its best gas mileage?

Answer: Take the derivative of m(v) and find it's critical points. The absolute maximum value will be the answers. The x value is the speed, while the corresponding y value is the gas mileage. Use m(v) to calculate the y value, not m'(v).

[math] m'(v) = 1.9e^{-v/40}(1-(v/40)) [/math]

x = 40 mi/hr

y = ~ 27.96 mi/gal

 

2. Let v(0) = 0 [mi/hr], and s(0) = 0 [mi]. Calculate the formulas for v(t) and s(t), given acceleration is constant and measured in mi/hr^2.

 

Answer: Use antiderivatives.

 

[math] \int a(t)*dt = v(t) = at [/math]

[math] \int v(t)*dt = \int at*dt = s(t) = (at^{2})/2 [/math]

 

3. If you plan to travel s = 8 miles at a constant acceleration a, show the time required.

 

Answer: Thus, s(t) = 8 miles, and solve for T.

[math] s(t) = (at^{2})/2[/math]

[math] T = \sqrt{16/a}[/math]

 

4. And I quote: "We now compute the total gasoline used in traveling 8 miles with a constant acceleration of a. Consider a small interval of time [t, t + delta t]. Explain why the distance traveled by the car in that interval is v*(delta time) = at*(delta time)"

Typed word for word, and this is where I am stuck. It could be something incredibly easy, because I seem to be missing what they are asking. I explained the second part by using units and canceling. What I don't understand is how I'm supposed to calculate the gas consumption when it depends on the speed the car is traveling. Since the acceleration is constant, the velocity will change, and I can't simply use my optimal velocity calculated in question one. Since the time is supposed to be a small interval, it could be a limits function, but I'm not sure if I'm on the right track. It has something to do with the last question, I'm sure.

 

Any thoughts? It would be highly appreciated. Thanks.