Question
For these items, you will attach work and a screenshot below. (5 points) Charles wants to go on a fishing trip. The price of the excursion varies based on the length of the trip. The line of best fit for these data is y=50x+75 , where the price of the trip is based on the number of hours of the trip. Show steps to solve this and attach below. If he paid $400, how many hours was his trip? Find the residual of the data point (2, 15.27) using the information in the graph below.
Answers
GPT-4o mini
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 \( x \):
1. Subtract 75 from both sides:
\[ 400 - 75 = 50x \]
\[ 325 = 50x \]
2. 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
1. Charles's trip was **6.5 hours** long.
2. 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.
### 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 \( x \):
1. Subtract 75 from both sides:
\[ 400 - 75 = 50x \]
\[ 325 = 50x \]
2. 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
1. Charles's trip was **6.5 hours** long.
2. 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.