To solve the two parts of the question involving Charles's fishing trip, let's break them down step by step.
Part 1: Finding Hours of the Trip
We have the equation of the line of best fit, which is given as: \[ y = 50x + 75 \] where \( y \) is the price of the trip and \( x \) is the number of hours.
Charles paid $400 for his trip. We need to find \( x \) (the number of hours) for which the price is $400.
Set \( y = 400 \): \[ 400 = 50x + 75 \]
Now, solve for \( x \):
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Subtract 75 from both sides: \[ 400 - 75 = 50x \] \[ 325 = 50x \]
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Divide both sides by 50: \[ x = \frac{325}{50} \] \[ x = 6.5 \]
Thus, Charles's trip lasted 6.5 hours.
Part 2: Finding the Residual
To find the residual for the data point (2, 15.27), we first need to determine the predicted value \( y \) when \( x = 2 \) using the line of best fit formula.
Substituting \( x = 2 \) into the equation \( y = 50x + 75 \): \[ y = 50(2) + 75 \] \[ y = 100 + 75 \] \[ y = 175 \]
Now, the actual data point is \( (2, 15.27) \), which means the actual price is $15.27 when \( x = 2 \).
The residual is calculated as: \[ \text{Residual} = \text{Actual} - \text{Predicted} \] Substituting the actual and predicted values: \[ \text{Residual} = 15.27 - 175 \] \[ \text{Residual} = -159.73 \]
Summary of Results
- Charles's trip was 6.5 hours long.
- The residual for the data point (2, 15.27) is -159.73.
Screenshot
Since I cannot provide a screenshot as per my capabilities, please ensure to take a screenshot of the completed calculations on your device or use an appropriate graphing tool to visualize the data if required.