Thanks for uploading the file! This appears to be an example of a water tank problem where we are calculating the maximum amount of water in the tank over a given time interval.

To summarize:

• The tank starts with 15 liters of water at time $t = 0$.
• Water is being pumped into the tank at a rate given by $N(t) = 17(0.69)^t$ (in liters per minute).
• Water is being drained from the tank at a rate given by $B(t) = 11(0.86)^t$ (in liters per minute).

The goal is to find the maximum amount of water in the tank from $t = 0$ to $t = 7$ minutes. The formula to calculate the amount of water in the tank at time $t$ is:

$A(t) = 15 + \int_0^t \left[ N(x) - B(x) \right] \, dx$

You are asked to find the critical points where the rate of change of water equals zero, which happens at $t \approx 1.9766$ minutes. From there, you can calculate the amount of water at $t = 0$, $t = 1.9766$, and $t = 7$.

The maximum amount of water is found to be 20.011 liters.

Would you like a more detailed explanation of any step, or do you have a specific question about this problem?

Here are some follow-up questions that might help:

1. How is the integral in $A(t)$ interpreted in this context?
2. Can you explain how to find the critical points for the function $A'(t)$?
3. How do you interpret the physical meaning of the maximum value of 20.011 liters?
4. What would happen if the rate of water in or out of the tank was changed?
5. How do you calculate the rate of change of the water level from $A'(t)$?

Tip: When finding critical points of a function involving integrals, it's often helpful to differentiate the integral and then solve for when the derivative equals zero.