I've seen questions like this before but they've given a time constraint. How do I do 12?
Are you an Extension student Oliver? Because I think this has strayed into that territory of difficulty
So we have this expression for cost in terms of speed,
per hour:
$$c=150+\frac{v^2}{80}$$
What we want is the
total cost for a 500km trip. To find how long THIS takes, we use our speed formula:
$$\text{Speed}=\frac{\text{Distance}}{\text{Time}}\\v=\frac{500}{t}\\\therefore t=\frac{500}{v}$$
So our trip takes that many hours. To find our
total cost, we multiply the cost per hour by the amount of hours taken that we just found.
$$C_\text{Total}=ct=\frac{500}{v}\left(50+\frac{v^2}{80}\right)$$
And now we're back to familiar territory, differentiate and find your minima! Does that help?