Let's consider the triangle formed by the spotlight, the closest point on the path to the spotlight, and the person. Let's call this closest point on the path "P", the person's position "A", and the spotlight "S".

We are given that the person is moving at a constant speed of 2 meters per second. Let's call the person's speed "v".

Let's consider the angle of elevation between the spotlight and the person. We'll call this angle "θ". We want to find how fast "θ" is changing with respect to time.

From the triangle, we can see that:
tan(θ) = PS / PA
PA is the distance between the person and the closest point on the path. We are given that PA = 30 meters.

To find PS, we can use the Pythagorean theorem:
PS^2 + PA^2 = SA^2
PS^2 + 40^2 = (PA + 40)^2
PS^2 + 1600 = (30 + 40)^2
PS^2 + 1600 = 70^2
PS^2 = 4900 - 1600
PS^2 = 3300
PS = sqrt(3300)
PS ≈ 57.45 meters

Now we can differentiate the equation tan(θ) = PS / PA with respect to time (t) using implicit differentiation.

d(tan(θ))/dt = d(PS/PA)/dt

Using the quotient rule, we have:

sec^2(θ) * dθ/dt = (dPS/dt * PA - PS * dPA/dt) / (PA^2)

We already know that dPS/dt is zero because the spotlight is not moving.

dθ/dt = (PA^2 * d(tan(θ))/dt) / (sec^2(θ) * PA - PS * dPA/dt)

To find d(tan(θ))/dt, we can differentiate tan(θ) with respect to PA using the chain rule:

d(tan(θ))/dt = sec^2(θ) * dθ/dt = (sec^2(θ) * dθ/dPA) * (dPA/dt)

We can solve for dθ/dPA by rearranging the equation tan(θ) = PS / PA to:

1 / cos^2(θ) * dθ/dPA = -PS / PA^2

Plugging this back into the equation for d(tan(θ))/dt:

d(tan(θ))/dt = (sec^2(θ) * -PS) / PA^2

Substituting this back into our expression for dθ/dt:

dθ/dt = (PA^2 * (sec^2(θ) * -PS) / PA^2) / (sec^2(θ) * PA - PS * dPA/dt)

Simplifying:

dθ/dt = (-PS) / (sec^2(θ) * PA - PS * dPA/dt)

Plugging in the values we know:

dθ/dt = (-57.45) / (sec^2(θ) * 30 - 57.45 * 2)

To find θ, we can use the inverse tangent function:

θ = tan^(-1)(PS / PA)
θ = tan^(-1)(57.45 / 30)

Plugging this value for θ back into the expression for dθ/dt:

dθ/dt ≈ (-57.45) / (sec^2(tan^(-1)(57.45 / 30)) * 30 - 57.45 * 2)

To find the numerical value of dθ/dt, we can calculate it using a calculator.

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