ANSWER:
- 0.01943 rad/sec
STEP-BY-STEP EXPLANATION:
The first thing is to make a drawing of what is mentioned in the statement, it would be the following:
Now, we have the following information:
[tex]\begin{gathered} \frac{dy}{dt}=15\text{ m/s} \\ x=50\text{ m} \\ y=190\text{ m} \end{gathered}[/tex]
In this right angle triangle formed by telescope of the boat, e can apply the tangent trigonometric ratio, like this:
[tex]\begin{gathered} \tan \theta=\frac{x}{y} \\ \text{ replacing} \\ \theta=\tan ^{-1}\mleft(\frac{x}{y}\mright) \end{gathered}[/tex]
Now, we implicitly derive with respect to t:
[tex]\begin{gathered} \frac{d}{dt}(\theta)=\frac{d}{dt}(\tan ^{-1}(\frac{x}{y})) \\ \frac{d}{dt}(\theta)=\frac{1}{1+(\frac{x}{y})^2}\cdot\frac{d}{dt}(\frac{x}{y}) \\ \frac{d}{dt}(\theta)=\frac{y^2}{x^2+y^2}\cdot x\cdot(-\frac{1}{y^2}\cdot\frac{dy}{dt}) \\ \frac{d}{dt}(\theta)=\frac{-x}{x^2+y^2}(\frac{dy}{dt}) \\ \text{ replacing} \\ \frac{d}{dt}(\theta)=\frac{-50}{50^2+190^2}\cdot(15) \\ \frac{d}{dt}(\theta)=-0.01943 \end{gathered}[/tex]
The angle of depression is changing at a rate of -0.01943 rad/sec when the boat is 190 m from the shore
