Question:
A lighthouse is located on an island 3 miles from the closest point on a straight shoreline. If the lighthouse light rotates clockwise at a constant rate of 9 revolutions per minute, how fast does the beam of light move towards the point on the shore closest to the island when it is 52 miles from that point
Answer:
The beam of light moves at [tex]16278\pi[/tex] miles/min
Step-by-step explanation:
This question is illustrated with the attached image
Taking the instructions in the question, one at a time.
A revolution of 9 per minute implies that:
[tex]\frac{d\theta}{dt} = \frac{9 * 2\pi\ rad}{1\ min}[/tex]
[tex]\frac{d\theta}{dt} = 18\pi \frac{rad}{min}[/tex]
Take tan of the angle in the attachment:
[tex]tan(\theta) =\frac{52}{3}[/tex]
Differentiate both sides with respect to time
[tex]\frac{d\ tan(\theta)}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
Rewrite as:
[tex]\frac{d\ tan(\theta)}{d\theta} * \frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
In calculus:
[tex]sec^2(\theta) =\frac{d\ tan(\theta)}{d\theta}[/tex] -- Chain rule
So:
[tex]sec^2(\theta) *\frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
In trigonometry:
[tex]sec^2(\theta) = tan^2(\theta) + 1[/tex]
So:
[tex](tan^2(\theta) + 1)\frac{d\theta}{dt} =\frac{52}{3} * \frac{dx}{dt}[/tex]
Recall that:
[tex]tan(\theta) =\frac{x}{3}[/tex]
[tex]((\frac{52}{3})^2 + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{52^2}{9} + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{2704}{9} + 1)\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{2704+9}{9})\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex](\frac{2713}{9})\frac{d\theta}{dt} =\frac{1}{3} * \frac{dx}{dt}[/tex]
Recall that: [tex]\frac{d\theta}{dt} = 18\pi \frac{rad}{min}[/tex]
[tex](\frac{2713}{9}) * 18\pi =\frac{1}{3} * \frac{dx}{dt}[/tex]
[tex]2713 * 2\pi =\frac{1}{3} * \frac{dx}{dt}[/tex]
Multiply both sides by 3
[tex]3 * 2713 * 2\pi =\frac{1}{3} * \frac{dx}{dt} * 3[/tex]
[tex]3 * 2713 * 2\pi =\frac{dx}{dt}[/tex]
[tex]16278\pi =\frac{dx}{dt}[/tex]
[tex]\frac{dx}{dt} = 16278\pi[/tex] miles/min
Hence:
The beam of light moves at [tex]16278\pi[/tex] miles/min
