1. Let z = x2 + y2, where x and y are increasing at the constant rates of 2 units per second and 3 units per second, respectively. What is the set of points for which the rate of change of z is zero?

 

A. {(x,y): 3y = -2x}

B. {(x,y): 3y = 2x}

C. {(x,y): x = y = 0}

D. {(x,y): 2y = -3x}

E. {(x,y): 2y = 3x}

 

 

Solution: The answer is A

 

The answer is obtained by solving for the rate of change of z with respect to time t and then equating this solution to zero. We begin with the given equation,

Take the derivative with respect to time t of both sides of the above equation, so that,

Since x and y are increasing at the constant rates of 2 units per second and 3 units per second respectively, the above equation becomes,

Now, simply set the above equation equal to zero and solve for the set of points (x,y).

So, the set of points for which the rate of change of z is zero is

{(x,y): 3y = -2x}