Exam-Style Question on Kinematics

The widest river in the world has a width of 11km at its widest point. Suppose there were a straight length of this river near Awkwardville (A) as shown in the diagram below. Points A and P lie on opposite banks, such that AP is the shortest distance across the river. Point B represents the centre of Bumblingburg which is located on the southern riverbank.

$$PB = 40km, AP = 11km \text{ and angle } A \hat{P}B = 90° $$

A boat travels at an average speed of \(12km h^{-1}\).
A bus travels along the straight road between P and B at an average speed of \(30kmh^{-1}\).

(a) Find the travel time, in hours, from A to B given that the boat is taken from A to P, and the bus from P to B.

(b) Find the travel time, in hours, from A to B given the boat travels directly to B.

There is a point D which lies on the road from P to B. such that \(BD = x km\).

(c) If the boat travels from A to D and the bus travels from D to B, find an expression, in terms of \(x\), for the travel time T, from A to B, passing through D.

(d) Find the value of \(x\) so that T is a minimum.

An excursion involves renting the boat and the bus. The cost to rent the boat is £50 per hour and the cost to rent the bus is £35 per hour.

(e) Find the new value of \(x\) so that the total cost to travel from A to B via D is a minimum.

(f) Write down the minimum total cost for this journey.