Questions: A lighthouse stands 400 m off a straight shore and the focused beam of its light revolves three times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 175 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse. Let x be the distance along the shore from point P to the beam and let θ be the angle through which the light revolves. Write an equation that relates x and θ. tan θ = x / 400 Differentiate both sides of the equation with respect to t. dx / dt = (400 sec^2 θ) dθ / dt When the beam strikes the point Q, its speed along the shore is about 8981 m / min.

A lighthouse stands 400 m off a straight shore and the focused beam of its light revolves three times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 175 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse.

Let x be the distance along the shore from point P to the beam and let θ be the angle through which the light revolves. Write an equation that relates x and θ.
tan θ = x / 400

Differentiate both sides of the equation with respect to t.
dx / dt = (400 sec^2 θ) dθ / dt

When the beam strikes the point Q, its speed along the shore is about 8981 m / min.

Transcript text: A lighthouse stands 400 m off a straight shore and the focused beam of its light revolves three times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 175 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse. Let x be the distance along the shore from point P to the beam and let $\theta$ be the angle through which the light revolves. Write an equation that relates x and $\theta$. \[ \tan \theta=\frac{x}{400} \] Differentiate both sides of the equation with respect to $t$. \[ \frac{d x}{d t}=\left(400 \sec ^{2} \theta\right) \frac{d \theta}{d t} \] When the beam strikes the point Q, its speed along the shore is about $8981 \mathrm{~m} / \mathrm{min}$.

Solution

Solution Steps

Step 1: Define Variables and Set Up the Problem

Let $ x $ be the distance along the shore from point P to the beam.
Let $ \theta $ be the angle through which the light revolves.
Given: Lighthouse height = 400 m, distance from P to Q = 175 m, and the light revolves 3 times per minute.

Step 2: Establish the Relationship

Use the tangent function: $ \tan(\theta) = \frac{400}{x} $.
Differentiate both sides with respect to $ t $: $ \sec^2(\theta) \frac{d\theta}{dt} = -\frac{400}{x^2} \frac{dx}{dt} $.

Step 3: Substitute Known Values and Solve

Given $ \frac{d\theta}{dt} = 6\pi $ radians per minute (since 3 revolutions per minute = $ 6\pi $ radians per minute).
At $ x = 175 $ m, $ \sec^2(\theta) = 1 + \tan^2(\theta) = 1 + \left(\frac{400}{175}\right)^2 = 1 + \left(\frac{16}{7}\right)^2 = 1 + \frac{256}{49} = \frac{305}{49} $.
Substitute into the differentiated equation: $ \frac{305}{49} \cdot 6\pi = -\frac{400}{175^2} \frac{dx}{dt} $.
Solve for $ \frac{dx}{dt} $: $ \frac{dx}{dt} = -\frac{305 \cdot 6\pi \cdot 175^2}{49 \cdot 400} $.