Quandaries and Queries


Name: Gina
Who is asking: Student
Level of the question: All

Question: I have the following word problem (related rates and motion) and cannot determine how to set it up: A point is moving on the graph of x3 + y2 = 1 in such a way that its y coordinate is always increasing at a rate of 2 units per second. At which point(s) is the x coordinate increasing at a rate of 1 unit per second. NOTE: The graphic icon beside the problem indicates that this is a calculator problem. However, I have no idea how to put the equation in and graph it. Any help will be GREATLY appreciated!



Hi Gina,

As the point moves along the curve both its x and y coordinated change with time. That is x and y are both functions of time t. For emphasis I might write x(t) for x and y(t) for y. The y coordinate is always increasing at a rate of 2 units per second, that is y'(t) = 2. You are asked to find x and y when x'(t) = 1 unit per second.

To find a relationship between x'(t) and y'(t) you can differentiate the original equation with respect to t. Thus

3 x2(t) x'(t) + 2 y(t) y'(t) = 0

But y'(t) is 2 for all values of t so

3 x2(t) x'(t) + 2 y(t) (2) = 0
3 x2(t) x'(t) + 4 y(t) = 0

You want to find x and y at the particular time when x'(t) = 1 so substitute x'(t) = 1 and get

3 x2(t) + 4 y(t) = 0

(At this point I would write the equations without the letter t as they looks a little simpler.) Hence you have two equations

x3 + y2 = 1 and
3 x2 + 4 y = 0

Can you solve these equations for x and y? This is maybe where you need your calculator.