Question

.A man rowing a boat away from a lighthouse 150 m high takes 2 minutes to change the angle of elevation of the top of lighthouse from 45° to 30°. Find the speed of the boat. (Use√3 = 1.732)

Answer 1

Given:-

• Length of lighthouse is 150 m.
• Angle of elevation is changing 45° to 30°.
• Time taken to change angle of elevation is 2 minutes.

To find:-

• Speed of boat.

Solution:-

Let, lighthouse be AB. BD be distance between lighthouse and Angle of elevation 45°. And, BC be the distance between lighthouse and Angle of elevation 30°.

Let, BD be x.

And DC be y.

In ∆ABD,

Tan 45° = AB/BD

$\longrightarrow \sf 1 = \dfrac{150}{x}$

• Cross multiple.

$\longrightarrow \sf x = 150$

x is 150 m.

BC = x + y = 150 + y

In ∆ABC,

Tan 30° = AB/BC

$\longrightarrow \sf \dfrac{1}{\sqrt{3}} = \dfrac{150}{150 + y}$

• Cross multiple.

$\longrightarrow \sf 150 + y = 150 \times \sqrt{3}$

$\longrightarrow \sf 150 + y = 150 \times 1.732$

$\longrightarrow \sf 150 + y = 259.8$

$\longrightarrow \sf y = 259.8 - 150$

$\longrightarrow \sf y = 109.8$

y is 109.8 m.

Conversion:-

Time = 2 min

1 min = 60 sec.

= 2 × 60

= 120

Time = 120 seconds

Speed = Distance/Time

= y/120

= 109.8/120

= 0.915

Therefore,

Speed of boat is 0.915 m/s²

Answer 2

Answer:

${\huge {\tt {\red {\underline {Solution}}}}}$

• Length of lighthouse = 150 m
• Angle of elevation is changing 45° to 30°.
• Time taken to change angle of elevation is 2 minutes.

$\huge \sf \underline \red {let}$

1. Lighthouse be AB.
2. BD be distance between lighthouse and Angle of elevation 45°.
3. BC be the distance between lighthouse and Angle of elevation 30°.

• BD = x
• DC = y

$\sf \: \tan(a) = \dfrac{ab}{bd}$

$\sf \: 1 = \dfrac{150}{x}$

$\sf \: x \: = 150 \times 1$

$\sf \: x \: = 150 \: m$

$\sf \: bc = x + y$

$\sf \: In \:  ∆ \: ABC$

$\sf \: \tan(30⁰) = \dfrac{ab}{bc}$

• Tan30⁰ = 1/√3

$\sf \dfrac{1}{ \sqrt{3} } = \dfrac{150}{150 + y}$

$\sf \implies \: 150 + y = 150 \times \sqrt{3}$

$\sf \implies \: 150 + y = 150 \times 1.732$

$\sf \implies150y = 259.8$

$\sf \implies \: y \: = 259.8 - 150$

$\sf \implies \: y \: = 109.8$

Conversion

Mintues to second

$\bf \: 2 \: min \: = 2 \times 60 = 120 \: sec$

Now finding speed

$\huge \bf \: speed \: = \frac{distance}{time}$

$\sf \implies \:speed = \dfrac{y}{120}$

$\sf \implies \: speed \: = \dfrac{109.8}{120}$

$\huge \fbox {\bf speed = 0.915}$