The Table Shows The Height Of Water In A Pool As It Is Being Filled.Height Of Water In A Pool$\[ \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Time \\ (min) \end{tabular} & \begin{tabular}{c} Height \\ (in.) \end{tabular} \\ \hline 2 & 8
Understanding the Problem
The given table displays the height of water in a pool over a period of time. The table consists of two columns: Time (in minutes) and Height (in inches). The data in the table is as follows:
| Time (min) | Height (in.) |
|---|---|
| 2 | 8 |
| 4 | 12 |
| 6 | 16 |
| 8 | 20 |
| 10 | 24 |
| 12 | 28 |
| 14 | 32 |
| 16 | 36 |
| 18 | 40 |
| 20 | 44 |
Analyzing the Data
The table shows a linear relationship between the time and the height of the water in the pool. As the time increases, the height of the water also increases at a constant rate. This indicates that the water is being filled at a constant rate.
Calculating the Rate of Filling
To calculate the rate of filling, we need to find the change in height over a given time period. Let's take the first two data points: (2, 8) and (4, 12).
The change in height is 12 - 8 = 4 inches. The change in time is 4 - 2 = 2 minutes.
The rate of filling is calculated as the change in height divided by the change in time:
Rate of filling = (change in height) / (change in time) = 4 / 2 = 2 inches per minute.
Finding the Equation of the Line
Since the table shows a linear relationship between the time and the height of the water, we can find the equation of the line using the slope-intercept form:
y = mx + b
where m is the slope (rate of filling) and b is the y-intercept (initial height).
We already know the slope (m) is 2 inches per minute. To find the y-intercept (b), we can use the first data point (2, 8):
8 = 2(2) + b 8 = 4 + b b = 4
Now that we have the slope (m) and the y-intercept (b), we can write the equation of the line:
y = 2x + 4
Using the Equation to Find the Height at a Given Time
We can use the equation of the line to find the height of the water at a given time. For example, let's find the height of the water at time t = 15 minutes:
y = 2x + 4 y = 2(15) + 4 y = 30 + 4 y = 34
Therefore, the height of the water at time t = 15 minutes is 34 inches.
Conclusion
In this article, we analyzed the given table to understand the relationship between the time and the height of the water in a pool. We calculated the rate of filling and found the equation of the line using the slope-intercept form. We also used the equation to find the height of the water at a given time. This problem demonstrates the importance of analyzing data and using mathematical concepts to solve real-world problems.
Mathematical Concepts Used
- Linear relationship
- Slope-intercept form
- Equation of a line
- Rate of filling
Real-World Applications
- Calculating the rate of filling of a pool or a tank
- Finding the height of water at a given time
- Analyzing data to understand the relationship between variables
Future Directions
- Using the equation of the line to find the height of the water at multiple times
- Analyzing the data to understand the effect of other variables on the rate of filling
- Using mathematical concepts to solve other real-world problems.
The Table Shows the Height of Water in a Pool as it is Being Filled: Q&A ====================================================================
Q: What is the rate of filling of the pool?
A: The rate of filling of the pool is 2 inches per minute. This means that for every minute that passes, the height of the water in the pool increases by 2 inches.
Q: How can I find the height of the water at a given time?
A: To find the height of the water at a given time, you can use the equation of the line: y = 2x + 4. Simply plug in the value of x (time) and solve for y (height).
Q: What is the initial height of the water in the pool?
A: The initial height of the water in the pool is 4 inches. This is the y-intercept of the equation of the line.
Q: Is the rate of filling constant?
A: Yes, the rate of filling is constant. This means that the height of the water in the pool increases at a constant rate of 2 inches per minute.
Q: Can I use this equation to find the height of the water at multiple times?
A: Yes, you can use this equation to find the height of the water at multiple times. Simply plug in the values of x (time) and solve for y (height).
Q: What are some real-world applications of this problem?
A: Some real-world applications of this problem include:
- Calculating the rate of filling of a pool or a tank
- Finding the height of water at a given time
- Analyzing data to understand the relationship between variables
Q: How can I analyze the data to understand the relationship between variables?
A: To analyze the data, you can use various statistical methods such as regression analysis, correlation analysis, and time series analysis. These methods can help you understand the relationship between the variables and make predictions about future values.
Q: What are some limitations of this problem?
A: Some limitations of this problem include:
- The data is limited to a small range of values
- The rate of filling is assumed to be constant
- The equation of the line is a simplification of the actual relationship between the variables
Q: How can I extend this problem to include more variables?
A: To extend this problem to include more variables, you can use techniques such as multiple regression analysis, which can help you understand the relationship between multiple variables.
Q: What are some future directions for this problem?
A: Some future directions for this problem include:
- Using the equation of the line to make predictions about future values
- Analyzing the data to understand the effect of other variables on the rate of filling
- Using mathematical concepts to solve other real-world problems.
Conclusion
In this Q&A article, we have discussed various aspects of the problem, including the rate of filling, the equation of the line, and real-world applications. We have also discussed some limitations and future directions for the problem. We hope that this article has provided a comprehensive understanding of the problem and its applications.