A tank holds 50 gal of water, which drains from a leak at...

A tank holds 50 gal of water, which drains from a leak at the bottom, causing the tank to empty in 20 min. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelli's Law gives the volume of water remaining in the tank after $t$ minutes as $$ V(t)=50\left(1-\frac{t}{20}\right)^{2} \quad 0 \leq t \leq 20 $$ a. Find $V(0)$ and $V(20)$ b. What do your answers to part (a) represent? c. Make a table of values of $V(t)$ for $t=0,5,10,15,20$ d. Find the net change in the volume $V$ as $t$ changes from 0 min to 20 min.

A tank holds 50 gal of water, which drains from a leak at the bottom, causing the tank to empty in 20 min. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelli's Law gives the volume of water remaining in the tank after $t$ minutes as
$$
V(t)=50\left(1-\frac{t}{20}\right)^{2} \quad 0 \leq t \leq 20
$$
a. Find $V(0)$ and $V(20)$
b. What do your answers to part (a) represent?
c. Make a table of values of $V(t)$ for $t=0,5,10,15,20$
d. Find the net change in the volume $V$ as $t$ changes from 0 min to 20 min.

Key Concepts

Torricelli's Law

Torricelli's Law is a principle in fluid dynamics that relates the speed of fluid exiting an opening to the height of the fluid above the opening. It stems from the conservation of energy and is used to model the rate at which liquid drains from a container, reflecting that the outflow velocity increases with the fluid’s depth due to higher pressure.

Function Evaluation

Function evaluation involves substituting specific input values into a function to obtain the corresponding output values. In the context of this problem, evaluating the volume function at key time points helps determine how the volume of water in the tank changes over time, identifying the initial and ending conditions.

Initial and Final Conditions

The concepts of initial and final conditions are crucial in understanding the behavior of dynamic systems. Evaluating the function at the start and end of the interval provides insight into the system’s state at these critical moments, which, in physical problems, often correspond to the full and empty states of a container.

Tabulating Function Values

Tabulating function values is a method of organizing computed outputs of a function at various inputs into a table form. This approach helps to clearly visualize the progression or trend of the function over an interval, making it easier to analyze patterns in the behavior of the system.

Net Change Calculation

Net change calculation involves finding the overall difference between the final and initial values of a function over a given interval. This concept is important in understanding the total effect or change in the system, such as the total volume of water lost from a tank over a specified time period.

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