question 14 a lighthouse is on an island 450 meters off shore. the lighthouse beam makes a full turn every 5…

Explanation:

Step1: Set up the relationship

Let $x$ be the distance along the shoreline from the point on the shore closest to the lighthouse and $\theta$ be the angle that the lighthouse - beam makes with the line perpendicular to the shore. We know that $\tan\theta=\frac{x}{450}$, where the distance from the lighthouse to the shore is $a = 450$ meters.

Step2: Differentiate with respect to time $t$

Differentiating both sides of $\tan\theta=\frac{x}{450}$ with respect to $t$ gives $\sec^{2}\theta\frac{d\theta}{dt}=\frac{1}{450}\frac{dx}{dt}$. The lighthouse beam makes a full - turn ($2\pi$ radians) every 5 seconds, so $\frac{d\theta}{dt}=\frac{2\pi}{5}$ rad/s. When $x = 1200$ meters, we first find $\sec\theta$. Since $\tan\theta=\frac{x}{450}=\frac{1200}{450}=\frac{8}{3}$, and $\sec^{2}\theta=1 + \tan^{2}\theta$, then $\sec^{2}\theta=1+\left(\frac{8}{3}\right)^{2}=1+\frac{64}{9}=\frac{9 + 64}{9}=\frac{73}{9}$.

Step3: Solve for $\frac{dx}{dt}$

Substitute $\sec^{2}\theta=\frac{73}{9}$ and $\frac{d\theta}{dt}=\frac{2\pi}{5}$ into $\sec^{2}\theta\frac{d\theta}{dt}=\frac{1}{450}\frac{dx}{dt}$. We have $\frac{73}{9}\times\frac{2\pi}{5}=\frac{1}{450}\frac{dx}{dt}$. Cross - multiply to get $\frac{dx}{dt}=\frac{73\times2\pi\times450}{9\times5}$. Simplify the right - hand side: $\frac{dx}{dt}=\frac{73\times2\pi\times450}{45}=73\times2\pi = 146\pi\approx458.676$ m/s.

Answer:

$458.676$