Final answer:
To solve for the rate at which the angle of elevation is changing as the helicopter descends, we use related rates calculus involving the tangent function and implicit differentiation with respect to time. The actual value for the rate of change would be found by performing the calculation after differentiating.
Step-by-step explanation:
The student is asking how fast the angle of elevation is changing as a helicopter is descending towards a tree stump, which is a problem involving related rates in calculus. To find the rate at which the angle of elevation θ is changing, we can apply the trigonometric relation between θ, the height of the helicopter above the stump (opposite side), and the horizontal distance from the observer to the stump (adjacent side).
Let's denote the height of the helicopter above the stump as y and the horizontal distance from the observer to the stump as x. Since x is constant at 9 m and y is decreasing, the angle of elevation θ can be expressed using the tangent function as θ = tan-1(y/x). As y changes, θ also changes. We can find the rate of change of θ using implicit differentiation with respect to time t.
By differentiating both sides with respect to t and plugging in the known values (y = 5.2 m, dy/dt = -18 m/s, and x = 9 m), we can solve for dθ/dt, the rate at which the angle of elevation is changing. The negative rate of dy/dt indicates that the helicopter is descending, hence the angle of elevation will be decreasing over time, meaning dθ/dt will be negative.
Based on the values given, and using the related rates formula, we can calculate the specific rate at which the angle of elevation is changing. The correct answer would have to be determined by performing the actual calculation, which involves differentiating the tangent function and substituting the given values.