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Verizon [17]
3 years ago
9

A lighthouse is located on a small island 5 km away from the nearest point P on a straight shoreline and its light makes seven r

evolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?

Mathematics
1 answer:
Vaselesa [24]3 years ago
6 0

Answer:

Light is moving at the speed = 77.78 km/min

Step-by-step explanation:

Given:

Distance between island and point P = 5 km

Moving along the the shoreline when it is 1 km from P.

So, x = 1

From the above statement light makes seven revolutions per minute.

Therefore,              

\frac{d\theta}{dt}=\frac{7\times 2\pi\ rad}{1\ min}  = 14\pi\ rad/min  

Solution:

From the given figure.

tan\theta=\frac{x}{5}

Substitute x = 1

tan\theta=\frac{1}{5}  -------------------(1)

Now, first we differentiate both side with respect to t.

\frac{d(tan\theta)}{dt} = \frac{d}{dt}.(\frac{x}{3})

Using chain rule.

\frac{d(tan\theta)}{dt}.\frac{d\theta}{dt}  = (\frac{1}{5})\frac{dx}{dt}

sec^{2} \theta.\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

(1+tan^{2} \theta).\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

Substitute tan\theta=\frac{1}{5} from equation 1.

(1+(\frac{1}{3})^{2} ).\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

(1+\frac{1}{9}).\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

Substitute \frac{d\theta}{dt}=14\pi

\frac{10}{9}.14\pi  = \frac{1}{5}.\frac{dx}{dt}

Applying cross multiplication rule.

\frac{dx}{dt}= \frac{700\pi }{9}

\frac{dx}{dt}= 77.78\pi

Therefore, light is moving along the shore at the speed of 77.78 km/min

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