If a cylindrical tank holds 100,000 gallons of water, which...

If a cylindrical tank holds $100,000$ gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $V$ of water remaining in the tank after t minutes as $$\mathrm{V}(\mathrm{t})=100,000\left(1-\frac{\mathrm{t}}{60}\right)^{2} \quad 0 \leqslant \mathrm{t} \leqslant 60$$ Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t. What are its units? For times $t=0,10,20,30,40,50,$ and 60 $\mathrm{min}$ , find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two.At what time is the flow rate the greatest? The least?

If a cylindrical tank holds $100,000$ gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $V$ of water remaining in the tank after t minutes as
$$\mathrm{V}(\mathrm{t})=100,000\left(1-\frac{\mathrm{t}}{60}\right)^{2} \quad 0 \leqslant \mathrm{t} \leqslant 60$$
Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t) as a function of t. What are its units? For times $t=0,10,20,30,40,50,$ and 60 $\mathrm{min}$ , find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two.At what time is the flow rate the greatest? The least?

Key Concepts

Function Evaluation in Modeling

Evaluating functions and their derivatives at specific points allows us to quantitatively assess the behavior of a model at different instances. This practice helps in summarizing and interpreting the performance of systems over time, determining key moments such as peak or minimal rates.

Units and Their Interpretation

Associating appropriate units with the rate of change is essential to ensure that the outcome is meaningful in its physical context. For example, if volume is measured in gallons and time in minutes, the derivative naturally carries the units of gallons per minute, which helps in understanding the actual flow rate.

Derivative

A derivative is the measure of how a function changes as its input changes, representing the instantaneous rate of change. In applied problems, it tells us how fast or slow a quantity is changing at a particular moment, which is essential in the analysis of dynamic systems.

Chain Rule

The chain rule is a fundamental differentiation technique used when one function is composed with another. It allows us to differentiate composite functions by taking the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

Instantaneous Rate of Change in Physical Models

In physical models, the instantaneous rate of change informs us how rapidly a physical quantity, such as volume or mass, is varying at a given time. This concept is critical for understanding processes like draining, filling, or cooling where rates change continuously.

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