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Exam-Style Question on KinematicsA mathematics exam-style question with a worked solution that can be revealed gradually |
Question id: 616. This question is similar to one that appeared on an IB AA Standard paper in 2021. The use of a calculator is allowed.
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.
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