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Exercise 121Calculus › Single-variable calculus › Differentiation › Optimisation and curve sketchingHigh schoolAdded 4 Nov 2026

The most economical speed of a truck

#optimisation#modelling#economics#cost per kilometre

Statement

A truck driver is paid 1818 € per hour. At speed vv (in km/h) the fuel costs 0.005 v20.005\,v^2 € per hour. At which speed is the cost per kilometre minimal? What is this cost?

Hint

Per hour the cost is 18+0.005v218+0.005v^2. To get the cost per kilometre, divide by the number of kilometres travelled per hour.

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Solution

Cost per km.
c(v)=18+0.005v2v=18v+0.005v,v>0.c(v)=\frac{18+0.005v^2}{v}=\frac{18}v+0.005v,\qquad v>0.

Derivative. c′(v)=−18v2+0.005c'(v)=-\dfrac{18}{v^2}+0.005, which vanishes when v2=3600v^2=3600, i.e. v=60v=60. It is negative on (0,60)(0,60) and positive on (60,∞)(60,\infty), so v=60v=60 is the global minimum on (0,∞)(0,\infty).

Answer. 6060 km/h, at a cost of c(60)=0.30+0.30=0.60c(60)=0.30+0.30=0.60 €/km.

Comparison. At 8080 km/h: c(80)=0.225+0.40=0.625c(80)=0.225+0.40=0.625 €/km, and at 4040 km/h: c(40)=0.45+0.20=0.65c(40)=0.45+0.20=0.65 €/km.

Remark. At the optimum the two parts of the cost are equal (0.300.30 each). This is typical of av+bv\frac av+bv: the minimum is where the two terms balance.