Solving Tricky Math Problems: Cylinders, Curves and Scooters

[myanmar]

I'm really having trouble with these three problems. I'd post my attempts but most of it is in graph from.

3. David subjects a cylindrical can to a certain transformation. During this transformation the radius and height vary continuously with time. The radius is increasing at 4 in/min, while the height is decreasing at 10 in/min. Is the volume of the can increasing or decreasing, and at what rate, when the radius is 3 inches and the height is 5 inches?
4. A particle moves along a path described by y=x^{2}. At what point along the curve are x and y changing at the same rate? Find this rate if at that time we have y = sin^{2} t and x = sin t
5. Two straight roads intersect at right angles near the Krupps factory at Baden-Baden. Fabio drives his scooter towards the intersection at a rate of 50 kph. Ilsa drives her scooter on the other road away from the intersection at a speed of 30 kph. When Fabio is 2 kilometers from the intersection and Ilsa is 4 kilometers from the intersection:
(a) How fast is the distance between them changing?
(b) Are Fabio and Ilsa getting closer together or farther apart?

 

[Pere Callahan]

Let's start with 3.
The question talks about cylindrical cans, heights, radii and volumes and all of these quantities change over time...oh my god

So my first question: Can you write down an equation for the height of the can after t minutes? For the radius of the can after t minutes?

Do you know how the volume of the can is related to the height and the radius and can you thus write down an expression for the volume of the can after t minutes?

You can assume that at the beginning ( that is after 0 minutes ) the height of the can is h0 and the radius is r0 ... what would be the volume of the can before the "transformation" starts?

 

[HallsofIvy]

Be careful how you understand Pere Callahan's suggestion. You don't actually have to write the volume as a function of time. You need to write a formula for the volume and use the chain rule to differentiate both sides with respect to t- which you can do even if there is no t explicitely in the formula.

What you do need to do is show what you have tried!