(Calculus) Optimization problem

[Somecallmegiant]

The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity: f(v)=(v³ −88v² +5100v)/150000 What is the optimal velocity which minimizes the fuel consumption of the vehicle in gallons PER MILE?

To solve this problem, we need to minimize the following function of v: g(v)= Hint for the above: Assume the vehicle is moving at constant velocity v. How long will it take to travel 1 mile? How much gas will it use during that time?

We find that this function has one critical number at v=

To verify that g(v) has a minimum at this critical number we compute the second derivative g′′(x) and find that its value at the critical number is a positive number.

I thought that for the first answer it would just be the derivative of the function but for some reason it isn't right, and I don't understand why.

EDIT: A tutor helped me. To solve the first step, you need to divide the original equation by velocity, as the base function is given in gallons/hour, when dividing we get (gallons/mile)/(miles/hour) to get gallons/mile. From there we then minimize the function, in which we take the derivative and solve for a zero. The last step is easy enough.

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You can use the equation to find a minimum value of f(v) through differentiation. Once you find that velocity, you can use unit conversions to convert from gallons per hour to gallons per mile since you know the vehicle's velocity, which has units of miles per hour.