To determine which equation represents the height of the parasail ([tex]\( y \)[/tex]) based on the speed of the motorboat ([tex]\( x \)[/tex]), we need to perform a linear regression analysis on the given data points. Here are the steps to solve this problem:
1. Collect the Data Points:
- Speed of the Motorboat (in miles per hour): [tex]\( [12.5, 14.5, 16.5, 18.5, 20.5, 22.5] \)[/tex]
- Height of the Parasail from the Water Surface (in feet): [tex]\( [5, 10, 15, 20, 25, 30] \)[/tex]
2. Perform Linear Regression:
We need to find the linear relationship [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Calculate the Slope ([tex]\( m \)[/tex]) and Intercept ([tex]\( b \)[/tex]):
- Given the data points and applying linear regression, the calculated slope and intercept are:
[tex]\[
m = 2.5
\][/tex]
[tex]\[
b = -26.25
\][/tex]
4. Formulate the Equation:
Using the slope and intercept we found, the equation of the height ([tex]\( y \)[/tex]) as a function of the speed ([tex]\( x \)[/tex]) is:
[tex]\[
y = 2.5x - 26.25
\][/tex]
5. Match the Equation with the Given Options:
Let's compare our derived equation [tex]\( y = 2.5x - 26.25 \)[/tex] with the given options:
- (A) [tex]\( y = 25x \)[/tex]
- (B) [tex]\( y = 0.4x \)[/tex]
- (C) [tex]\( y = 25x - 26.25 \)[/tex]
- (D) [tex]\( y = 0.4x + 4.2 \)[/tex]
- (E) [tex]\( y = 0.4x - 4.2 \)[/tex]
None of the given options exactly match our derived equation [tex]\( y = 2.5x - 26.25 \)[/tex]. Therefore, none of the provided options are correct based on our analysis.
6. Conclusion:
The correct equation that represents the height of the parasail based on the speed of the motorboat is:
[tex]\[
y = 2.5x - 26.25
\][/tex]
But since none of the options match, there is no correct answer from the provided choices.
