The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism ABCDEF. The base ACFD of the roof is a horizontal rectangle, and the cross-
section ABC of the roof is an isosceles triangle with AB = BC.
The lengths of AC and CF are 2x cm and ycm respectively, and the height of BE above the base of
the roof is rcm.
The total surface area of the five faces of the roof is 600 cm and the volume of the roof is V cm.
(a) Show that v = kx(300 - x?), where k = Va+b and a and b are integers to be determined. [6]
(b) Use differentiation to determine the value of x for which the volume of the roof is a maximum.
(c) Find the maximum volume of the roof. Give your answer in cm', correct to the nearest integer.
[1]
(d) Explain why, for this roof, x must be less than a certain value, which you should state.
[2]​