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 € per hour. At speed (in km/h) the fuel costs € per hour. At which speed is the cost per kilometre minimal? What is this cost?
Hint
Per hour the cost is . To get the cost per kilometre, divide by the number of kilometres travelled per hour.
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Solution
Cost per km.
Derivative. , which vanishes when , i.e. . It is negative on and positive on , so is the global minimum on .
Answer. km/h, at a cost of €/km.
Comparison. At km/h: €/km, and at km/h: €/km.
Remark. At the optimum the two parts of the cost are equal ( each). This is typical of : the minimum is where the two terms balance.