Answer:
[tex]\frac{650\pi}{7}miles/min[/tex]
Step-by-step explanation:
We are given that
Distance of lighthouse from closest point P=7 miles
[tex]\frac{d\theta}{dt}=5 rev /min=5\times 2\pi rad/min=10\pi rad/min[/tex]
[tex]tan\theta=\frac{Perpendicular\;side}{base}[/tex]
Using the formula
[tex]tan\theta=\frac{x}{7}[/tex]
Differentiate w.r.t time
[tex]sec^2\theta \frac{d\theta}{dt}=\frac{1}{7}\frac{dx}{dt}[/tex]
Using the formula
[tex]\frac{d(tan\theta)}{d\theta}=sec^2\theta[/tex]
[tex](1+tan^2\theta)\frac{d\theta}{dt}=\frac{1}{7}\frac{dx}{dt}[/tex]
x=4 miles
Substitute the value
[tex](1+(\frac{4}{7})^2)\times 10\pi=\frac{1}{7}\frac{dx}{dt}[/tex]
[tex](1+\frac{16}{49})70\pi=\frac{dx}{dt}[/tex]
[tex]\frac{65\times 70\pi}{49}=\frac{dx}{dt}[/tex]
[tex]\frac{dx}{dt}=\frac{650\pi}{7}miles/min[/tex]
