Date | May 2019 | Marks available | 6 | Reference code | 19M.1.AHL.TZ1.H_5 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 1 |
Command term | Find | Question number | H_5 | Adapted from | N/A |
Question
A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at t = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.
A car travels along the road at a speed of 24 ms−1. Let the position of the car be X and let OĈX = θ.
Find dθdt, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
let OX = x
METHOD 1
dxdt=24 (or −24) (A1)
dθdt=dxdt×dθdx (M1)
3tanθ=x A1
EITHER
3sec2θ=dxdθ A1
dθdt=243sec2θ
attempt to substitute for θ=0 into their differential equation M1
OR
θ=arctan(x3)
dθdx=13×11+x29 A1
dθdt=24×13(1+x29)
attempt to substitute for x=0 into their differential equation M1
THEN
dθdt=243=8 (rad s−1) A1
Note: Accept −8 rad s−1.
METHOD 2
dxdt=24 (or −24) (A1)
3tanθ=x A1
attempt to differentiate implicitly with respect to t M1
3sec2θ×dθdt=dxdt A1
dθdt=243sec2θ
attempt to substitute for θ=0 into their differential equation M1
dθdt=243=8 (rad s−1) A1
Note: Accept −8 rad s−1.
Note: Can be done by consideration of CX, use of Pythagoras.
METHOD 3
let the position of the car be at time t be d−24t from O (A1)
tanθ=d−24t3(=d3−8t) M1
Note: For tanθ=24t3 award A0M1 and follow through.
EITHER
attempt to differentiate implicitly with respect to t M1
sec2θdθdt=−8 A1
attempt to substitute for θ=0 into their differential equation M1
OR
θ=arctan(d3−8t) M1
dθdt=81+(d3−8t)2 A1
at O, t=d24 A1
THEN
dθdt=−8 A1
[6 marks]