Science:Math Exam Resources/Courses/MATH104/December 2012/Question 05 (b)/Solution 2
The diagram from part a has been recopied for ease of reading
Proceeding via the hints, we are interested in solving for . We first note that triangles BBob and triangle BPL are similar (they share an angle and both contain a 90 degree angle). Thus, we have that
- .
Multiplying by 30, reducing and differentiating with respect to time gives
- .
To solve for we will need , the speed at which Bob's shadow tip is changing. To do this we consider that the distance between the two tips is decreasing by 11ft/s and so
- .
To determine we turn to the similar triangle problem for Alice. Triangles AAlice and triangle APL are similar (they share an angle and both contain a 90 degree angle). Thus, we have that
Multiplying by 15, reducing and differentiating with respect to time gives
We are given that and so we have that
- .
Therefore we have that
and finally then that
and so Bob is walking towards the lamppost at 3ft/s. Notice the sign we get makes sense because we are told that each person is walking towards the lamp in the question!