Let's walk through the problem step by step.

Given:

The volume of water remaining in the tank at time $t$ is modeled by: $V(t) = 70 \left( 1 - \frac{t}{20} \right)^2$ where $t$ is the time in minutes, and $0 \leq t \leq 20$.

Part (a)

You've already calculated:

• $V(0) = 70$ gallons, which represents the initial volume of water in the tank (i.e., at $t = 0$).
• $V(20) = 0$ gallons, representing the volume of water at the end (i.e., after 20 minutes when the tank is empty).

Part (b) - Table of values

Let's calculate the values of $V(t)$ for $t = 0, 5, 10, 15, 20$.

$V(t) = 70 \left( 1 - \frac{t}{20} \right)^2$

1. At $t = 0$: $V(0) = 70 \left( 1 - \frac{0}{20} \right)^2 = 70 \times 1^2 = 70 \, \text{gallons}$
2. At $t = 5$: $V(5) = 70 \left( 1 - \frac{5}{20} \right)^2 = 70 \left( 1 - 0.25 \right)^2 = 70 \times 0.75^2 = 70 \times 0.5625 = 39.375 \, \text{gallons}$
3. At $t = 10$: $V(10) = 70 \left( 1 - \frac{10}{20} \right)^2 = 70 \left( 1 - 0.5 \right)^2 = 70 \times 0.5^2 = 70 \times 0.25 = 17.5 \, \text{gallons}$
4. At $t = 15$: $V(15) = 70 \left( 1 - \frac{15}{20} \right)^2 = 70 \left( 1 - 0.75 \right)^2 = 70 \times 0.25^2 = 70 \times 0.0625 = 4.375 \, \text{gallons}$
5. At $t = 20$: $V(20) = 70 \left( 1 - \frac{20}{20} \right)^2 = 70 \times 0^2 = 0 \, \text{gallons}$

So the correct table of values is:

$t$ (minutes)	$V(t)$ (gallons)
0	70
5	39.375
10	17.5
15	4.375
20	0

Part (c) - Net change in volume from $t = 0$ to $t = 10$

The net change in the volume of water in the tank is given by: $\text{Net change} = V(0) - V(10)$ From the calculations in Part (b):

• $V(0) = 70$ gallons
• $V(10) = 17.5$ gallons

Thus, the net change is: $\text{Net change} = 70 - 17.5 = 52.5 \, \text{gallons}$

Final Answer for Part (c):

The net change in the volume of water in the tank when $t$ changes from 0 to 10 is 52.5 gallons.

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Would you like more details or have any questions?

Here are 5 relative questions to expand on this topic:

1. How does Torricelli's Law relate to real-world fluid dynamics?
2. How can we calculate the rate of water flow at any given time $t$?
3. How would the equation change if the tank had a different initial volume?
4. Can you derive the time at which the tank is half empty using this model?
5. What happens if there are multiple leaks, how would the model change?

Tip: When analyzing changes in volume or rates, understanding the physical context (like pressure in this case) helps connect mathematical models to real-world situations.