A tank holds 100 gallons of water which drains from a leak at the bottom, causing the tank to empty in 40 minutes. Toricelli's Law gives the volume F(x) of water remaining in the tank, X in gallons, after x minutes by the function F(x)=100 1-- (1. 40
A tank holds 100 gallons of water which drains from a leak at the bottom, causing the tank to empty in 40 minutes. Toricelli's Law gives the volume F(x) of water remaining in the tank, X in gallons, after x minutes by the function F(x)=100 1-- (1. 40
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Transcribed Image Text:**c) Find an equation for the inverse function \( F^{-1}(x) \) if you only use the inputs described in a).**
![### Understanding Torricelli's Law for Water Drainage
In this educational segment, we will explore the application of Torricelli's Law to understand how water drains from a tank. Consider a tank initially holding 100 gallons of water. A leak at the bottom causes the tank to empty over a period of 40 minutes. Torricelli's Law gives the volume \( F(x) \) of water remaining in the tank, in gallons, after \( x \) minutes. The function describing this process is given by:
\[
F(x) = 100 \left( 1 - \frac{x}{40} \right)^2
\]
### Explanation of the Function
- **Initial Volume**: The tank starts with 100 gallons of water.
- **Variable \( x \)**: Represents the time in minutes since the leakage started.
- **Term \(\left( 1 - \frac{x}{40} \right)\)**: This term represents the proportion of time elapsed relative to the total time (40 minutes) needed to empty the tank.
- **Equation Breakdown**:
- When \( x = 0 \) (at the start), \( F(0) = 100 \left( 1 - \frac{0}{40} \right)^2 = 100 \) gallons.
- When \( x = 40 \) (at the end), \( F(40) = 100 \left( 1 - \frac{40}{40} \right)^2 = 0 \) gallons.
By substituting different values for \( x \) in the formula, you can find out how much water remains in the tank at any given minute.
### Summary
This function reveals the non-linear nature of water drainage through a leak, emphasizing how the rate of drainage accelerates as the tank empties. Through understanding this equation, students can appreciate fundamental principles of fluid dynamics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98c4113f-b35f-4b8e-9a05-0e521af0c658%2F6e58e5b0-afd7-4a5d-becc-c2520d7c7933%2Fy8ica1n_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Torricelli's Law for Water Drainage
In this educational segment, we will explore the application of Torricelli's Law to understand how water drains from a tank. Consider a tank initially holding 100 gallons of water. A leak at the bottom causes the tank to empty over a period of 40 minutes. Torricelli's Law gives the volume \( F(x) \) of water remaining in the tank, in gallons, after \( x \) minutes. The function describing this process is given by:
\[
F(x) = 100 \left( 1 - \frac{x}{40} \right)^2
\]
### Explanation of the Function
- **Initial Volume**: The tank starts with 100 gallons of water.
- **Variable \( x \)**: Represents the time in minutes since the leakage started.
- **Term \(\left( 1 - \frac{x}{40} \right)\)**: This term represents the proportion of time elapsed relative to the total time (40 minutes) needed to empty the tank.
- **Equation Breakdown**:
- When \( x = 0 \) (at the start), \( F(0) = 100 \left( 1 - \frac{0}{40} \right)^2 = 100 \) gallons.
- When \( x = 40 \) (at the end), \( F(40) = 100 \left( 1 - \frac{40}{40} \right)^2 = 0 \) gallons.
By substituting different values for \( x \) in the formula, you can find out how much water remains in the tank at any given minute.
### Summary
This function reveals the non-linear nature of water drainage through a leak, emphasizing how the rate of drainage accelerates as the tank empties. Through understanding this equation, students can appreciate fundamental principles of fluid dynamics.
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