A-lighthouse-L-is-located-on-a-small-island-2-km-from-the-nearest-point-A-on-a-long-the-straigh-shoreline-If-the-lighthouse-lamp-rotates-at-3-revolutions-per-minute-how-fast-is-the-illuminated-s

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Question Number 132225 by bemath last updated on 12/Feb/21

$$$$$$\to \mathrm{A}\mathrm{lighthouse}\mathrm{L}\mathrm{is}\mathrm{located}\mathrm{on}\mathrm{a}$$$$\mathrm{small}\mathrm{island}2\mathrm{km}\mathrm{from}\mathrm{the}$$$$\mathrm{nearest}\mathrm{point}\mathrm{A}\mathrm{on}\mathrm{a}\mathrm{long}\mathrm{the}$$$$\mathrm{straigh}\mathrm{shoreline}.\mathrm{If}\mathrm{the}$$$$\mathrm{lighthouse}\mathrm{lamp}\mathrm{rotates}\mathrm{at}3$$$$\mathrm{revolutions}\mathrm{per}\mathrm{minute}.\mathrm{how}\mathrm{fast}$$$$\mathrm{is}\mathrm{the}\mathrm{illuminated}\mathrm{spot}\mathrm{P}\mathrm{on}\mathrm{the}$$$$\mathrm{shoreline}\mathrm{moving}\mathrm{along}\mathrm{the}$$$$\mathrm{shoreline}\mathrm{when}\mathrm{it}\mathrm{is}4\mathrm{km}\mathrm{from}\mathrm{A}$$

Answered by mr W last updated on 12/Feb/21

Commented by mr W last updated on 12/Feb/21

$$x=2\tan \theta$$$$\frac{dx}{dt}=\frac{dx}{d\theta }\times \frac{d\theta }{dt}=\frac{2}{{\cos }^2\theta }\times \frac{d\theta }{dt}=2(1+{\tan }^2\theta )\times \frac{d\theta }{dt}$$$$=2[1+{\left(\frac{x}{2}\right)}^2]\times \frac{d\theta }{dt}$$$$atx=4:$$$$\frac{dx}{dt}=2[1+{\left(\frac{4}{2}\right)}^2]\times \frac{3\times 2\pi }{60}=\pi =3.141km/s$$

Commented by bemath last updated on 12/Feb/21

$$\mathrm{i}\mathrm{got}\frac{\mathrm{dx}}{\mathrm{dt}}{\mid }_{\mathrm{x}=4\mathrm{km}}=60\pi \mathrm{km}/\min$$$$\mathrm{sir}.\mathrm{correct}?$$$$60\pi \approx 188.495559\mathrm{km}/\min$$$$$$

Commented by mr W last updated on 12/Feb/21

$$correct!$$

Commented by bemath last updated on 12/Feb/21

$$\mathrm{yes}..\mathrm{thanks}!$$

Commented by otchereabdullai@gmail.com last updated on 13/Feb/21

$$\mathrm{nice}\mathrm{one}\mathrm{prof}$$

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