Answer :
To solve the problem of determining which sinusoidal function models the tide height data correctly, we need to compare both given functions with the actual heights measured at different times. Here’s a step-by-step way to understand and solve the problem:
1. Understanding the Functions:
We have two potential sinusoidal functions:
- [tex]\( y = -5.5 \sin(-0.5x + 13.1) + 35.8 \)[/tex]
- [tex]\( y = -0.5 \sin(-5.5x + 13.1) + 35.8 \)[/tex]
2. Evaluating the Functions:
You should evaluate each function at specific x-values (in this case, we can assume x as a sequence of time points). Since we have five data points, consider x to be 0, 1, 2, 3, and 4, representing different times:
3. Modeling Data Points:
For each function, compute the modeled heights for x = 0, 1, 2, 3, and 4.
4. Calculating Differences:
Compare these modeled heights with the actual measurements given: 19, 12, 9.5, 18.5, and 7. Calculate the absolute differences between the modeled heights and actual heights for each time point.
5. Summing Differences:
Sum the absolute differences for each function to get a total difference value for each model.
6. Determining the Best Model:
Compare the total differences. The function with the smaller total difference more accurately models the data (i.e., the sum of differences indicates which function's modeled values are closer to the measured values).
Conclusion:
The function that has the smallest total difference is the one that better fits the data. This is done to determine how well each mathematical model approximates the actual recorded tide heights. By following these steps, you'll be able to decide which sinusoidal function is the best representation for the data.
1. Understanding the Functions:
We have two potential sinusoidal functions:
- [tex]\( y = -5.5 \sin(-0.5x + 13.1) + 35.8 \)[/tex]
- [tex]\( y = -0.5 \sin(-5.5x + 13.1) + 35.8 \)[/tex]
2. Evaluating the Functions:
You should evaluate each function at specific x-values (in this case, we can assume x as a sequence of time points). Since we have five data points, consider x to be 0, 1, 2, 3, and 4, representing different times:
3. Modeling Data Points:
For each function, compute the modeled heights for x = 0, 1, 2, 3, and 4.
4. Calculating Differences:
Compare these modeled heights with the actual measurements given: 19, 12, 9.5, 18.5, and 7. Calculate the absolute differences between the modeled heights and actual heights for each time point.
5. Summing Differences:
Sum the absolute differences for each function to get a total difference value for each model.
6. Determining the Best Model:
Compare the total differences. The function with the smaller total difference more accurately models the data (i.e., the sum of differences indicates which function's modeled values are closer to the measured values).
Conclusion:
The function that has the smallest total difference is the one that better fits the data. This is done to determine how well each mathematical model approximates the actual recorded tide heights. By following these steps, you'll be able to decide which sinusoidal function is the best representation for the data.
