2023 TWO(e)

Excellence
Question
A police helicopter is flying above a straight horizontal section of motorway chasing a speeding
car.

The helicopter is flying at a constant speed of 72 m s−172\ \mathrm{m\ s^{-1}} and at a constant height of 400400 metres
above the ground. The helicopter is attempting to catch up with the car.

When the direct distance from the helicopter to the car is 25002500 metres, the angle of depression, θ\theta,
between the horizontal and the line of sight from the helicopter to the car is increasing at a rate of
0.002 rad s−10.002\ \mathrm{rad\ s^{-1}}.
Loading diagram...
Calculate the speed of the car at this instant.

*You must use calculus and show any derivatives that you need to find when solving this problem.*
Official Answer
Loading diagram...
Let xx = horizontal distance between the helicopter
and the car.
Let yy = direct distance between the helicopter and
the car.

Given: dθdt=0.002 rad s−1\dfrac{d\theta}{dt}=0.002\,\mathrm{rad\,s^{-1}}

tan⁡θ=400x\tan\theta=\dfrac{400}{x}

x=400cot⁡θx=400\cot\theta

dxdθ=−400cosec⁡2θ\dfrac{dx}{d\theta}=-400\cosec^2\theta

=−400sin⁡2θ=-\dfrac{400}{\sin^2\theta}

dxdt=dxdθ×dθdt\dfrac{dx}{dt}=\dfrac{dx}{d\theta}\times\dfrac{d\theta}{dt}

=−400sin⁡2θ×0.002=-\dfrac{400}{\sin^2\theta}\times0.002

=−0.8sin⁡2θ=-\dfrac{0.8}{\sin^2\theta}

When y=2500y=2500, sin⁡θ=4002500\sin\theta=\dfrac{400}{2500}

θ=0.1607 rad\theta=0.1607\,\mathrm{rad}

dxdt=−0.8sin⁡2(0.1607)\dfrac{dx}{dt}=\dfrac{-0.8}{\sin^2(0.1607)}

=31.25=31.25

When the helicopter is travelling at 72 m s−172\,\mathrm{m\,s^{-1}},
The speed of the car =72−31.25=72-31.25

=40.75 m s−1=40.75\,\mathrm{m\,s^{-1}}

(=146.7 km/hr)(=146.7\,\mathrm{km/hr})
Grading Criteria

Achievement (u)

  • Finds dxdθ\dfrac{dx}{d\theta}.

Merit (r)

  • Finds an expression for dxdt\dfrac{dx}{dt}.

Excellence T1

  • T1
    Finds the value for dxdt=−31.25\dfrac{dx}{dt}=-31.25

    With correct derivatives.

    OR
    Finds correct solution but with one minor error.

Excellence T2

  • T2
    Finds dxdt=−31.25\dfrac{dx}{dt}=-31.25

    with correct derivatives.

    AND
    The speed of the car =40.76 m s−1=40.76\,m\,s^{-1}.
Video Explanation
2023 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 42:04)
Subscribe
Exam Paper (2023)Assessment Schedule (2023)
Related Questions
2024 THREE(e)Excellence

The diagram below shows part of the graph of the function $f(x) = e^{-x^2}$, where $x \geq 0$. [DI...

2024 THREE(d)Merit

Jamie is doing some baking and pouring the flour to form a conical pile. The height o...

2024 TWO(e)Excellence

The graph of the function $y = \dfrac{xe^{3x}}{2x + k}$, where $k$ is a non-zero constant, has a sin...

2024 TWO(c)Merit

Show that $y = \sin(x^2) - \cos(x)$ is a solution to the equation $\dfrac{d^2y}{dx^2} + 4x^2\,y = 2...

2024 TWO(a)Achievement

A function is defined parametrically by the pair of equations: $x = 3t^2 + 1$ and $y = \cos t.$ Fi...