A tank holds 70 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 20 minutes. The tank drains faster when it is nearly full, because the pressure on the leak is greater. Torricelli's Law models the volume V (in gallons) of water remaining in the tank after t minutes as
V(t) = 70

1 − 
t
20
	2
 
,  0 ≤ t ≤ 20.

(a) Find V(0).
V(0) = 
70
 
Correct: Your answer is correct.


What does this value represent?
The volume of water in the tank after 
0
 
Correct: Your answer is correct.
 min is 
70
 
Correct: Your answer is correct.
 gal.

Find V(20).
V(20) = 
0
 
Correct: Your answer is correct.


What does this value represent?
The volume of water in the tank after 
20
 
Correct: Your answer is correct.
 min is 
0
 
Correct: Your answer is correct.
 gal.

(b) Make a table of values of V(t) for t = 0, 5, 10, 15, 20.
t	V(t)
0	
70
 
Correct: Your answer is correct.
5	
65
 
Incorrect: Your answer is incorrect.
10	
52
 
Incorrect: Your answer is incorrect.
15	
30
 
Incorrect: Your answer is incorrect.
20	
0
 
Correct: Your answer is correct.

(c) Find the net change of the volume of water in the tank when t changes from 0 to 10.
17.5
 
Incorrect: Your answer is incorrect.
 gal


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**.

---

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.

A tank holds 70 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 20 minutes. The tank drains faster when it is nearly full, because the pressure on the leak is greater. Torricelli's Law models the volume V (in gallons) of water remaining in the tank after t minutes as V(t) = 70(1 - t/20)^2, 0 ≤ t ≤ 20. (a) Find V(0). (b) Make a table of values of V(t) for t = 0, 5, 10, 15, 20. (c) Find the net change of the volume of water in the tank when t changes from 0 to 10.

Learn how to solve a water drainage problem modeled by Torricelli's Law: V(t) = 70(1 - t/20)^2. This problem explores how water drains from a tank over time and covers algebraic functions and real-world fluid dynamics. Suitable for students in Grades 11-12.

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