At the big New Year's Eve gala, a giant crystal ball descends at midnight at a rate of 3 meters per minutes. A party-goer stands and watches from 5 meters away (horizontally).

Question

Question a. Find the rate of change of the (diagonal) distance between the party-goer and the crystal ball when the crystal ball is 12 meters off the ground.

Question b. Find the rate of change of the angle elevation from the party-goer to the crystal ball when the crystal ball is 12 meters off the ground.

Answer 1

You are trying to find dD/dt and dθ/dt from dy/dt, y, and x D is distance along hypotenuse and y is the height of the chandelier. The problem should state that the chandelier is 12 ft above PG's eye level, not the ground.

D = (x²+y²)^1/2

θ = tan^-1(y/x)

Now, you just take the derivatives D and θ with t as well as x and y (chain rule). Since x is constant dx/dt = 0

dD/dt = 1/2 (x² +y²)^-1/2 2y dy/dt

dθ/dt = 1/(1+(y/x)²) * (1/x) dy/dt

dy/dt is - 3m/min. x = 5 m y = 12 m

Please check over the results before plugging in.