Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>


b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?

Verified step by step guidance

1. Identify the relationship between the height of the falcon and the angle θ. Use trigonometric functions to express this relationship, noting that tan(θ) = opposite/adjacent.
2. Set up a right triangle where the height of the falcon above the ground is the opposite side and the horizontal distance from the base of the cliff to the falcon is the adjacent side.
3. Differentiate the relationship you established in step 1 with respect to time to find dθ/dt, applying implicit differentiation as necessary.
4. Use the given height of the falcon (60 ft) to find the corresponding horizontal distance using the tangent function.
5. Substitute the values into the differentiated equation to solve for the rate of change of θ with respect to the bird’s height.