# Calculus/Related Rates/Solutions

< Calculus | Related Rates

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1. A spherical balloon is inflated at a rate of . Assuming the rate of inflation remains constant, how fast is the radius of the balloon increasing at the instant the radius is ?

Known:

Take the time derivative:

Solve for :

Plug in known values:

2. Water is pumped from a cone shaped reservoir (the vertex is pointed down) in diameter and deep at a constant rate of . How fast is the water level falling when the depth of the water is ?

Known:

Take the time derivative:

Solve for :

Plug in known values:

3. A boat is pulled into a dock via a rope with one end attached to the bow of a boat and the other wound around a winch that is in diameter. If the winch turns at a constant rate of , how fast is the boat moving toward the dock?

Let be the number of revolutions made and be the distance the boat has moved toward the dock.

Known:

(each revolution adds one circumferance of distance to s)

Solve for :

Take the time derivative:

Plug in known values:

4. At time a pump begins filling a cylindrical reservoir with radius 1 meter at a rate of cubic meters per second. At what time is the liquid height increasing at 0.001 meters per second?