Related Rates: A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes two revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? (Round your answer to one decimal place.)
the light turns through angle θ where dθ/dt = 4π.
If the light hits x km away from P, then
x = 4tanθ
So,
knowing that tanθ = 1/4, plug in the numbers and find dx/dt:
dx/dt = 4 sec^2(θ) dθ/dt
To solve this related rates problem, we can use the concept of similar triangles. Let's break down the problem and identify the quantities involved:
1. The distance from the lighthouse to the nearest point on the shoreline is 4 km.
2. The beam of light revolves around the lighthouse twice per minute.
We are interested in finding the rate at which the beam of light is moving along the shoreline when it is 1 km from the nearest point on the shoreline.
To solve this problem, follow these steps:
Step 1: Identify the known and unknown quantities.
- Known quantities: The distance from the lighthouse to the nearest point on the shoreline (4 km) and the rate at which the light makes two revolutions per minute.
- Unknown quantity: The rate at which the beam of light is moving along the shoreline when it is 1 km from the nearest point.
Step 2: Set up a diagram and label the relevant variables.
Draw a diagram to represent the situation. Label the lighthouse as L, the nearest point on the shoreline as P, and the position of the beam of light as A. Let PA represent the distance we are interested in (1 km).
L_____________________A
/
/
/
P
Step 3: Establish a relationship between the known and unknown quantities.
Since we are interested in the rate of change of distance PA with respect to time, we can establish a relationship using similar triangles.
The distance from the lighthouse to the beam of light can be represented as LA. Notice that triangle LPA and triangle LAA' are similar, where A' is a point along the shoreline at a distance being covered by the beam of light.
Therefore, we can establish the following relationship:
(LA)/(LP) = (AA')/(AP)
Step 4: Differentiate the relationship with respect to time.
Differentiate both sides of the equation with respect to time (t) to find the rate of change.
d(LA)/dt = d[(AA')/(AP)]/dt
Step 5: Substitute all known values and solve for the unknown rate.
We know the following:
- LA = 1 km (the distance we are interested in)
- LP = 4 km
- d(LA)/dt is the rate we want to find
- d[(AA')/(AP)]/dt is the rate at which the beam of light moves along the shoreline when it is 1 km from the nearest point.
Plugging in the known values into the equation, we have:
1/(4) = d[(AA')]/dt / d(AP)/dt
Step 6: Solve for the unknown rate.
Simplifying the equation, we get:
d[(AA')]/dt = (1/4) * d(AP)/dt
Substituting the known value of d(AP)/dt, which is twice the rate at which the light revolves per minute (2 revolutions per minute), we have:
d[(AA')]/dt = (1/4) * 2 revolutions per minute = 0.5 revolutions per minute
Therefore, the rate at which the beam of light is moving along the shoreline when it is 1 km from the nearest point P is 0.5 revolutions per minute.
Remember to convert this rate to a linear speed if necessary.