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).
Derivatives are used to find maximum profit.