Question

I am stuck on these problems. I have tried it multiple times and gotten it wrong.
1. A plane flies horizontally at an altitude of 3 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/6, this angle is decreasing at a rate of π/4 rad/min. How fast is the plane traveling at that time?
2. Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor h = 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 1 ft/s. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q? (Round your answer to two decimal places.)
3. Gravel is being dumped from a conveyor belt at a rate of 25 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 6 ft high? (Round your answer to two decimal places.)

Answer 1

Too bad you didn't show your work.
#1. Let x be the distance of the plane from the point directly overhead. Then
3/x = tanθ
so, x = 3cotθ
dx/dt = -3csc^2θ dθ/dt
when θ = π/6, x = 3√3 and cscθ = 2
So, the plane's speed,
dx/dt = -3(2)(-π/4)

#2. At the moment in question, AP = 13, so BP = 26.
Let x = BQ, y=AQ. So we know that AP+BP=39, and
√(y^2+12^2) + √(x^2+12^2) = 39
y/√(y^2+144) dy/dt + x/√(x^2+144) dx/dt = 0
Plugging our numbers,
5/13 * 1 + √133/26 dx/dt = 0

#3. If the base of the cone has radius r, then the height h=2r, so the volume
v = 1/3 πr^2 h = 1/3 π (h/2)^2 h = 1/12 πh^3
dv/dt = 1/4 π h^2 dh/dt
Plugging in the numbers,
25 = π/12 * 36 * dh/dt