Given the equation y = 2x + 1, where both x and y are functions of t, how do you find dy/dt when dx/dt = 15 and x = 4? Also, how do you find dx/dt when dy/dt = 2 and x = 40?

To solve the problem, we will use the relationship between the variables x and y, defined by the equation y = 2x + 1.

We begin with the derivative of y with respect to t:

y = 2x + 1

Differentiate both sides with respect to t:

dy/dt = 2(dx/dt)

Now, we substitute dx/dt = 15 into the equation:

dy/dt = 2(15) = 30

Therefore, when x = 4 and dx/dt = 15, it follows that dy/dt = 30.

We again refer back to our derivative expression:

dy/dt = 2(dx/dt)

This time we are given dy/dt = 2. We can set up our equation:

2 = 2(dx/dt)

To find dx/dt, divide both sides by 2:

dx/dt = 1

So, when x = 40 and dy/dt = 2, it follows that dx/dt = 1.

This method demonstrates how to find the rates of change for x and y based on their relationship as defined by the equation.