Question

1. the diagram shows two marathon runners, a and b, heading towards the finish line of a race. from an apartment window 80 metres above the ground and 20 metres behind the finish line, tony measures the angle of depression of the runners to be 28° and 24°, respectively. a) calculate the distance between the runners to the nearest metre. b) a is travelling at a constant speed of 4.5 m/s, while b is travelling at a constant speed of 5.1 m/s. which runner will finish the race first?

Explanation:

Step1: Use tangent function for distance calculation

Let the distance of runner A from the building base be $x_A$ and for runner B be $x_B$. We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For runner A with $\theta_A = 28^{\circ}$ and height of window $h = 80$m, $\tan28^{\circ}=\frac{80}{x_A}$, so $x_A=\frac{80}{\tan28^{\circ}}$. For runner B with $\theta_B=24^{\circ}$, $\tan24^{\circ}=\frac{80}{x_B}$, so $x_B = \frac{80}{\tan24^{\circ}}$.

Step2: Calculate the distance between the runners

The distance $d$ between the runners is $d=\vert x_B - x_A\vert$.
$x_A=\frac{80}{\tan28^{\circ}}\approx\frac{80}{0.5317}\approx150.46$m.
$x_B=\frac{80}{\tan24^{\circ}}\approx\frac{80}{0.4452}\approx179.70$m.
$d=x_B - x_A=179.70 - 150.46 = 29.24\approx29$m.

Answer:

29 m