Consider the illustration, which shows a rotating beam of light located 0.5 mile from a shoreline. The beam rotates at a rate of 4 revolutions per minute. How fast (in miles per minute) is the distance between the beam and the point where it strikes the shore changing at the instant when x = .25 miles?
It depends upon the angle that the shore makes with the beam.
What is x ?
An illustration is needed for me to make sense of this question.
I recall seeing this type of question before, and I will assume it is the standard type.
Usually we are to find the speed of the light as it moves along the shore.
Draw a perpendicular line from the lighthouse to the shore.
Let the angle between the line to the shore and the beam of light be Ø , and let the light rotate counterclockwise.
dØ/dt = 4 (2π) radians/minute = 8π rad/min
Let the beam of light be x miles along the shore
tanØ = x/.5
x = .5tanØ
dx/dt = .5 sec^2 Ø dØ/dt
when x = .25
tanØ = .25/.5 = 1/2
cosØ = 2/√5
secØ = √5/2
sec^2 Ø = 5/4
dx = .5(5/4)(8π) miles/min = 5π miles/minute
To find the rate at which the distance between the beam and the point where it strikes the shore is changing, we can use the concept of related rates.
Let's denote the distance between the beam and the point where it strikes the shore as y, and the angle between the beam and the shoreline as x.
From the given information, we know that the beam rotates at a rate of 4 revolutions per minute. This means that the angle x is changing at a constant rate of 4 revolutions per minute.
We are interested in finding how fast the distance y is changing with respect to time when x = 0.25 miles.
To solve this problem, we need to find an equation that relates y and x. From the given information, we can see that the triangle formed by the beam, the shoreline, and the distance y is a right triangle.
Using this right triangle, we can apply the Pythagorean theorem:
y^2 + (0.5 miles)^2 = x^2
Differentiating both sides of the equation with respect to time:
2y * (dy/dt) = 2x * (dx/dt)
We are given that dx/dt (the rate at which x is changing) is 4 revolutions per minute.
At the instant when x = 0.25 miles, we can find y by substituting this value in the equation:
y^2 + (0.5 miles)^2 = (0.25 miles)^2
Simplifying the equation:
y^2 = (0.25 miles)^2 - (0.5 miles)^2
y^2 = 0.0625 miles^2 - 0.25 miles^2
y^2 = -0.1875 miles^2
Since y represents a distance, it cannot be negative. Hence, y must be zero at this instant.
Now, we can substitute the values into the derived equation:
2(0) * (dy/dt) = 0.25 miles * (4 revolutions per minute)
Simplifying:
0 = 1 mile per minute * (dy/dt)
Therefore, the rate at which the distance between the beam and the point where it strikes the shore is changing is 0 miles per minute at the instant when x = 0.25 miles.
To solve this problem, we can apply the concepts of trigonometry and rates of change.
First, let's define some variables:
- Let R be the distance between the beam and the point where it strikes the shore.
- Let x be the angle (in radians) that the beam has rotated.
- Let t be the time in minutes.
Now, we need to find an equation that relates the variables R, x, and t. We know that the beam rotates at a rate of 4 revolutions per minute, which means it completes 4 full circles in 1 minute. In terms of radians, this is equivalent to 4 * 2π radians.
Since the beam is rotating at a constant rate, we can express the angle x as a function of time t using the equation:
x = (4 * 2π) * t
Next, we need to find a relationship between R and x. From the problem statement, when the beam is located 0.5 miles from the shoreline (R = 0.5), the angle x is 0. To find a general equation for R as a function of x, we'll use some trigonometry.
Consider a right triangle, where the hypotenuse is the radius of the rotation circle (R) and the adjacent side is the distance from the beam to the point where it strikes the shore. The opposite side represents the distance between the beam and the point on the shore. Since the angle x is the central angle of the rotating beam, we have:
cos(x) = R / 0.5
R = 0.5 * cos(x)
Now, we can differentiate both sides of the equation with respect to time t:
dR/dt = d(0.5 * cos(x)) / dt
To find the value of dR/dt when x = 0.25 miles, we need to substitute this value into the equation above. However, we can express x in terms of t using the equation x = (4 * 2π) * t. Thus:
dR/dt = d(0.5 * cos((4 * 2π) * t)) / dt
To calculate the value of dR/dt when t = t0, we differentiate the equation with respect to t and substitute t = t0 into the resulting expression.
After substituting the value of x and differentiating, we can calculate dR/dt at the desired instant when x = 0.25 miles.
Please note that exact calculations are needed to determine the numerical value of dR/dt at that instant.