Question

A man on a cliff observes a boat at an angle of depression of 30° which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is found to be 60°. Find the total time taken by the boat to reach the shore.

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Answer 1

Answer:

To Find:-

• Tíme taken by boàt to reach the shorè.
• Which concépt is beíng usèd?

$\begin{gathered}\pink{\Large{\underbrace{\underline{\bf{GIVEN\::}}}}} \\ \end{gathered}$

A man on a clíff observes a boàt at an angle of depressíon of 30°, which is approaching the shorè to the point immediately benéath the observer with a uniform speed.

Síx minutes later the angle of depressíon of the boàt is found to be 60°

$\red{\Large{\underbrace{\underline{\bf{SOLUTION\::}}}}}$

☛ First assume the two positíons of the boàt at the two ínstants are A & D.

Let,

The speed of the boàt is v m/min.

h is the height of the clíff.

C is the locatíon of the man.

[Note ➝ The figure is given as in the attachment.]

✳ The distance covered from the point A to the point D is given as,

$\begin{gathered}\orange\bigstar\:\mid\:\bf\purple{Distance\:=\: Speed\times{Time}\:}\:\mid\:\green\bigstar \\ \end{gathered}$

➟ Distance (AD) = v × 6

➟ Distance (AD) = 6v

☛ Now, assume that the boàt takes t time to reach the shorè then the distance covered from the point D to the point B is given as,

➟ Distance (DB) = v × t

➟ Distance (DB) = vt

☛ Now apply the trigonometric ratio in the ∆DBC,

$\Large\purple\star⋆ \begin{gathered}\mid\:\bf\pink{\tan\theta\:=\:\dfrac{Perpendicular}{Base}\:}\:\mid \\ \end{gathered}$

$➵ \begin{gathered}\sf{\tan{60}\:=\:\dfrac{BC}{DB}\:} \\ \end{gathered}$

$➵ \begin{gathered}\sf{\sqrt{3}\:=\:\dfrac{h}{vt}\:} \\ \end{gathered}$

$➵ \begin{gathered}\sf{h\:=\:\sqrt{3}\:vt}---(1) \\ \end{gathered}$

☛ Now apply the trigonometric ratio in the ∆ABC,

$➵ \begin{gathered}\sf{\tan{30}\:=\:\dfrac{BC}{AB}\:} \\ \end{gathered}$

$➵ \begin{gathered}\sf{\dfrac{1}{\sqrt{3}}\:=\:\dfrac{h}{6v\:+\:vt}\:} \\ \end{gathered}$

$➵ \begin{gathered}\sf{h\:=\:\dfrac{v\:(6\:+\:t)}{\sqrt{3}}\:}---(2) \\ \end{gathered}$

☛ Compare the value of h from the equation (1) & equation (2),

$\begin{gathered}:\implies\:\sf{\sqrt{3}\:vt\:=\:\dfrac{v\:(6\:+\:t)}{\sqrt{3}}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\sf{\sqrt{3}t\:=\:\dfrac{6\:+\:t}{\sqrt{3}}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\sf{\sqrt{3}t\times{\sqrt{3}}\:=\:6\:+\:t\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\sf{3t\:=\:6\:+\:t\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\sf{3t\:-\:t\:=\:6\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\sf{2t\:=\:6\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\sf{t\:=\:\dfrac{6}{2}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\bf{t\:=\:3\: minutes} \\ \end{gathered}$

✰ Therefore, the boàt will take 3 minutes to reach the shorè.

☛ As given that, take 6 minutes to reach the point D from the point A and we have find that the boàt takes 3 minutes to reach the shorè from the point D, thus the total time taken is,

⇒ Total time = (6 + 3) minutes

⇒ Total time = 9 minutes

∴ ⑴ The boàt will take 9 minutes to reach the shorè.

Answer 2

Answer refers in the attachment.

$\sf$

Let the time taken to cover DA be t minutes. Then, ratio of distances = ratio of times taken to cover them.

So, 2 : 1 = 6 : t

$\sf \implies \frac{2}{1} = \frac{6}{t}$

$\sf \implies 2t = 6$

$\sf \implies t = 3$

So, the time taken by the boat to cover DA is 3 minutes.

Total time taken by the boat to reach the shore

= (6 + 3) minutes

= 9 minutes

$\sf$