Answer:
[tex]\dfrac{800 \pi}{7} \ miles/min[/tex]
Step-by-step explanation:
From the given information:
We learned that the light rotates 8 revolutions per minute.
Therefore;
[tex]\dfrac{d \theta}{d t}= \dfrac{8 * 2 \pi}{1 \ min}[/tex]
[tex]\dfrac{d \theta}{d t}= 16\pi \ rads/min\\[/tex]
From the trigonometry formula, the tanget of the lighthouse in relation to the closest point on a straight shoreline can be expressed as:
[tex]tan \theta = \dfrac{x}{7}[/tex]
By differentiation with respect to t; we have:
[tex]\dfrac{d(tan \theta)}{dt} = \dfrac{1}{7} * \dfrac{dx}{dt}[/tex]
[tex]\dfrac{d(tan \theta)}{d \theta}* \dfrac{d \theta}{dt} = \dfrac{1}{7} * \dfrac{dx}{dt}[/tex]
By applying the chain rule:
[tex]sec^2 * \dfrac{d \theta }{d t} = \dfrac{1}{7} * \dfrac{dx}{dt}[/tex]
[tex](1 + tan ^2 \theta) \dfrac{d \theta }{dt} = \dfrac{1}{7} * \dfrac{dx}{dt}[/tex]
When x = 3 ; we have [tex]tan \theta = \dfrac{1}{7}[/tex]
[tex]\bigg (1 + \dfrac{1}{49} \bigg ) \dfrac{d \theta }{dt} = \dfrac{1}{7} * \dfrac{dx}{dt}[/tex]
Also;
[tex]\dfrac{d \theta}{d t}= 16\pi \ rads/min\\[/tex]
[tex]\bigg (1 + \dfrac{1}{49} \bigg ) 16 \pi = \dfrac{1}{7} * \dfrac{dx}{dt}[/tex]
[tex]\bigg ( \dfrac{50}{7} \bigg ) 16 \pi = \dfrac{dx}{dt}[/tex]
[tex]\dfrac{dx}{dt} = \dfrac{800 \pi}{7} \ miles/min[/tex]
