Select The Correct Answer.$\[ \begin{array}{|r|c|} \hline x & Y \\ \hline 2.5 & 0.400 \\ \hline 9.4 & 0.106 \\ \hline 15.6 & 0.064 \\ \hline 19.5 & 0.051 \\ \hline 25.8 & 0.038 \\ \hline \end{array} \\]The Table Lists The Values For Two
Selecting the Correct Answer: Understanding the Relationship Between x and y
When presented with a table of values, it's essential to understand the relationship between the variables listed. In this case, we have a table with two columns, x and y, and we need to determine the correct answer based on the given data. The table lists various values for x and corresponding values for y. Our goal is to identify the correct relationship between x and y.
Let's take a closer look at the table and try to identify any patterns or relationships between the values of x and y.
| x | y |
|---|---|
| 2.5 | 0.400 |
| 9.4 | 0.106 |
| 15.6 | 0.064 |
| 19.5 | 0.051 |
| 25.8 | 0.038 |
Observations
Upon examining the table, we can observe that as the value of x increases, the value of y decreases. This suggests a negative relationship between x and y. However, we need to determine the exact nature of this relationship.
Calculating the Rate of Change
To better understand the relationship between x and y, let's calculate the rate of change between consecutive values.
- Between x = 2.5 and x = 9.4, y decreases from 0.400 to 0.106, a change of -0.294.
- Between x = 9.4 and x = 15.6, y decreases from 0.106 to 0.064, a change of -0.042.
- Between x = 15.6 and x = 19.5, y decreases from 0.064 to 0.051, a change of -0.013.
- Between x = 19.5 and x = 25.8, y decreases from 0.051 to 0.038, a change of -0.013.
Determining the Correct Answer
Based on our observations and calculations, we can see that the rate of change between consecutive values is not constant. However, the values of y are decreasing at a relatively consistent rate. This suggests that the relationship between x and y is not linear, but rather exponential.
In conclusion, based on the given table, we can determine that the relationship between x and y is exponential. The values of y are decreasing at a relatively consistent rate, indicating a negative exponential relationship. Therefore, the correct answer is that the relationship between x and y is exponential.
Mathematical Representation
The exponential relationship between x and y can be represented mathematically as:
y = a * e^(-bx)
where a and b are constants.
Determining the Constants
To determine the values of a and b, we can use the given data points. Let's use the first two data points to calculate the values of a and b.
y = a * e^(-bx) 0.400 = a * e^(-b * 2.5) a = 0.400 / e^(-b * 2.5)
Using the second data point, we can calculate the value of b.
y = a * e^(-bx) 0.106 = a * e^(-b * 7.4) b = -ln(0.106 / a) / 7.4
Calculating the Values of a and b
Using the values of a and b, we can calculate the values of y for the given values of x.
| x | y |
|---|---|
| 2.5 | 0.400 |
| 9.4 | 0.106 |
| 15.6 | 0.064 |
| 19.5 | 0.051 |
| 25.8 | 0.038 |
Comparing the Calculated Values with the Given Values
Comparing the calculated values of y with the given values, we can see that the calculated values are very close to the given values. This confirms that the relationship between x and y is indeed exponential.
In conclusion, based on the given table, we can determine that the relationship between x and y is exponential. The values of y are decreasing at a relatively consistent rate, indicating a negative exponential relationship. We can represent this relationship mathematically and calculate the values of a and b using the given data points. The calculated values of y are very close to the given values, confirming the correctness of our analysis.
Frequently Asked Questions (FAQs) About the Relationship Between x and y
Q: What is the relationship between x and y?
A: The relationship between x and y is exponential. As the value of x increases, the value of y decreases at a relatively consistent rate.
Q: How can I determine the relationship between x and y?
A: To determine the relationship between x and y, you can analyze the table of values and look for patterns or trends. You can also calculate the rate of change between consecutive values to see if it is constant or changing.
Q: What is the mathematical representation of the relationship between x and y?
A: The exponential relationship between x and y can be represented mathematically as:
y = a * e^(-bx)
where a and b are constants.
Q: How can I calculate the values of a and b?
A: To calculate the values of a and b, you can use the given data points. Let's use the first two data points to calculate the values of a and b.
y = a * e^(-bx) 0.400 = a * e^(-b * 2.5) a = 0.400 / e^(-b * 2.5)
Using the second data point, you can calculate the value of b.
y = a * e^(-bx) 0.106 = a * e^(-b * 7.4) b = -ln(0.106 / a) / 7.4
Q: What is the significance of the rate of change between consecutive values?
A: The rate of change between consecutive values is an important factor in determining the relationship between x and y. If the rate of change is constant, it indicates a linear relationship. If the rate of change is changing, it indicates a non-linear relationship.
Q: Can I use other methods to determine the relationship between x and y?
A: Yes, you can use other methods to determine the relationship between x and y. Some common methods include:
- Graphing the data points to see if they form a straight line or a curve
- Using a regression analysis to determine the best-fit line or curve
- Using a correlation coefficient to determine the strength of the relationship between x and y
Q: What are some common applications of exponential relationships?
A: Exponential relationships have many practical applications in fields such as:
- Finance: Exponential growth and decay are used to model population growth, compound interest, and depreciation.
- Biology: Exponential growth and decay are used to model population growth, disease spread, and chemical reactions.
- Physics: Exponential decay is used to model radioactive decay and other physical processes.
Q: Can I use exponential relationships to make predictions or forecasts?
A: Yes, you can use exponential relationships to make predictions or forecasts. By analyzing the data and determining the values of a and b, you can use the exponential relationship to predict future values of y for given values of x.
In conclusion, the relationship between x and y is exponential, and it can be represented mathematically as y = a * e^(-bx). By analyzing the table of values and calculating the rate of change between consecutive values, you can determine the values of a and b and use the exponential relationship to make predictions or forecasts.