Dear Math Central,

Can you please help me with this question:

The cost of running a car at an average speed of V km/h is given by c= 100 + (V2 / 75) cents per hour. Find the average speed (to the nearest km/h) at which the cost of a 1000 km trip is a minimum.

I got the equation D = ST so 1000 = Vt and tried substituting this into c and deriving dc/dt but failed when i tried to equate dc/dt = 0. Am I doing this wrong?

Much thanks and appreciation,

Edward (Yr 12 student)


Hi Edward.

The wording of the question tells us that the cost of the trip is based on a single speed - that which is the cheapest to run the car at. So you have an equation expressing the cost (C) of running the car based on its speed (V):

C= 100 + (V2 / 75)

At what point is C a minimum? To find this, calculate the first derivative, set that equal to zero and solve for V. This won't tell you what the actual cost is (that's only one more step if you want to do it), but the question is just asking for the speed anyway.

The total cost/distance calculations D = ST and 1000 = Vt are right, but the question isn't asking about those, so they're irrelevant.

Hope this helps,
Stephen La Rocque.

PS: Setting the derivative equal to zero helps you find minimums and maximums, but you normally need to do a bit more work to see whether what you are looking at is a minimum or a maximum. In the case of your question though, it's pretty clear that the cost function is a parabola opening upwards, so the only possible critical point is a minimum - and that's what you want.