The linear velocity of the beam along the shore, when the angle with the shoreline is 45 degrees, is 96π miles per minute, determined by relating x and θ through trigonometric ratios and angular velocity calculations.
We need to find the relationship between x and θ. Using the trigonometric ratio of tangent, we have tan θ = x/3. Solving for x, we get x = 3 tan θ.
We need to find the angular velocity of the beam, which is the rate of change of θ with respect to time. Since the beam makes 8 revolutions per minute, we can convert this to radians per minute by multiplying by 2π. The angular velocity is ω = 8 * 2π = 16π radians per minute.
We need to find the linear velocity of the beam along the shore, which is the rate of change of x with respect to time. Using the chain rule, we have dx/dt = (dx/dθ) * (dθ/dt). To find dx/dθ, we can use the quotient rule on x = 3 tan θ. We get dx/dθ = 3 sec^2 θ. Substituting the values of dx/dθ and dθ/dt, we get dx/dt = 3 sec^2 θ * 16π = 48π sec^2 θ miles per minute.
We need to evaluate the linear velocity at the instant when θ = 45 degrees. Using the trigonometric identity of secant, we have sec^2 45 = 2. Substituting this into the expression for dx/dt, we get dx/dt = 48π * 2 = 96π miles per minute. This is the velocity of the beam of light along the shore when it makes an angle of 45 degrees with the shoreline.