1) Air is leaking out of a spherical balloon at the rate of 3 cubic cm/sec. When the radius is 5 cm, how fast is the surface area decreasing?
2) A cone shaped paper cup is being filled with water at the rate of 3 cubic cm/sec. The height of the cup is 10 cm and the radius of its base is 5 cm. How fast is the water level rising when the level is 4 cm?
3) A ship B is moving westward toward a fixed point A at a speed of 12km/h. At the moment when the ship is 72 km away from A, a ship C passes through A, heading due south at 10km/h. What is the rate of change in the distance between the ships two hours after ship C passed through A?
Well to start we have to calculate the distance Ship C has traveled from A, the new distance Ship B is away from A, and the distance between the two.
New distance for Ship B =
New distance for Ship C =
Distance between the two =
Now we can use the Pythagorean theorem to calculate the rate of change between the two ships:
Well we know the rate of change for the two ships:
(this is negative because Ship B is going towards A, making the distance smaller)
And we plug in what we know:
Then you solve for dc which is the rate of change between the two ships .