Question

Two variable quantities A and B are found to be related by the equation given below. What is the rate of change dA/dt at the moment when A = 3 and dB/dt = 5?

A3+83=35
dA /dt=__ when A = 3 and dB / dt = 5. (Simplify your answer.)

Answer 1

Explanation:

To find the rate of change dA/dt, we need to take the derivative of the equation A³ + 8A = 35 with respect to time.

First, let's rearrange the equation to isolate A³:

A³ = 35 - 8A

Now, we can take the derivative of both sides with respect to time (t):

d/dt(A³) = d/dt(35 - 8A)

Using the chain rule, the left side becomes:

3A² * dA/dt

And the right side becomes:

0 - 8 * dA/dt

So, we have the equation:

3A² * dA/dt = -8 * dA/dt

Now, let's substitute the given values A = 3 and dB/dt = 5:

3(3²) * dA/dt = -8 * 5

Simplifying, we have:

27 * dA/dt = -40

Finally, we can solve for dA/dt:

dA/dt = -40 / 27

So, the rate of change dA/dt is approximately -1.481 when A = 3 and dB/dt = 5.