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Derivatives and Maximum Profit

3. Parcel Post

A firm wishes to use the services of a parcel delivery company to transport a cylindrical package. The package has volume V=pr2l, where l is the length of the package in metres and r is the radius of the circular end in metres. The parcel delivery company will only transport parcels provided that the sum of the length and diameter of a parcel is not more than 2 metres. The firm wishes to maximise the volume of the package.

(a) Explain why l=2-2rif we want to maximize V.

(b) Show that the quantity to be maximised is V=2pr2-2pr3.

(c) Calculate Dv/dr and d2V/dr2

(d) Find the stationary points of V and determine their nature. Hence state the maximum volume of the cylinder (correct to 2 decimal places).

(e) Sketch the graph of V as a function of r.

Solution Summary

Derivatives are used to find maximum profit.