Question

A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards him. If it takes 12 minutes for the angle of depression to change from 30 degree to 45 degree, how soon after this, will the car reach the tower? Give your answer to the nearest second.

Answer 1

Hope it helps..........

Answer 2

Answer: The car will reach the tower in 16 minutes approximately.

Step-by-step explanation:

Since we have given that

Time taken for the angle of depression to change from 30° to 45° = 12 minutes

As shown in the figure.

BC = x units

So, In ΔABC, we have

$\tan 45^\circ=\frac{AB}{BC}\\\\1=\frac{AB}{BC}\\\\AB=BC=x\ units$

We will keep CD = 12 minutes in minutes as equality nullify the unit.

So, there is no need to change the units.

Consider, ΔABD,

Let Distance of CD = y units

$\\tan 30^\circ=\frac{AB}{BD}\\\\\frac{1}{\sqrt{3}}=\frac{x}{x+y}\\\\and\\\\Speed=\frac{y}{12}=\frac{x}{t}\\\\\frac{y}{x}=\frac{12}{t}$

$\tan 30^\circ=\frac{AB}{BD}\\\\\frac{1}{\sqrt{3}}=\frac{x}{x+y}\\\\\sqrt{3}=\frac{x+y}{x}\\\\\sqrt{3}-1=\frac{y}{x}\\\\\frac{12}{t}=\sqrt{3}-1\\\\t=\frac{12(\sqrt{3}+1}{(\sqrt{3}-1)(\sqrt{3}+1)}\\\\t=\frac{12(\sqrt{3}+1}{2}\\\\t=6(\sqrt{3}+1)\\\\t=16.39\\\\x\approx 16\ minutes$

Hence, The car will reach the tower in 16 minutes approximately.