4912: A particle moves

Question

A particle moves along the curve y = 3x2 + 1 in such a way that the y value is increasing at the rate of 3 units per second. At what rate is x changing when x = 2?

• Increasing 1/3 unit/sec
• Increasing 1/4 unit/sec
• Decreasing 1/3 unit/sec
• Decreasing 1/4 unit/sec

Answer 1

*Note, since y is positively related to x (as x increases, y increases) we know that x will be increasing.

 

Now, to solve this problem, just find the rate of change of x with respect to time as:

dy/dt = d/dt (3x² + 1) = 6x * dx/dt (From chain rule):

 

dx/dt = (1/6x)*dy/dt

when x = 2, we have:

dx/dt = (1/12)*3 = 1/4 units per second.

 

*Note that this rate is positive, thus x is increasing, which matches intuition.