Questions: Suppose the line y=2.8333 x-22.4967 describes the relation between the club-head speed (in miles per hour), x and the distance a golf ball travels (in yards), y. (a) Predict the distance a golf ball will travel if the club-head speed is 100 mph. (b) Suppose the observed distance a golf ball traveled when the club-head speed was 100 mph was 265 yards. What is the residual? (a) The golf ball will travel yards. (Round to the nearest tenth as needed.)

Solution Steps

To solve this problem, we will use the given linear equation to predict the distance a golf ball will travel for a given club-head speed. For part (a), substitute the given club-head speed into the equation to find the predicted distance. For part (b), calculate the residual by subtracting the predicted distance from the observed distance.

To predict the distance $y$ a golf ball will travel for a club-head speed $x = 100$ mph, we use the linear equation:

$$y = 2.8333x - 22.4967$$

Substituting $x = 100$:

$$y = 2.8333(100) - 22.4967 = 260.8333$$

Rounding to the nearest tenth, we find:

$$y \approx 260.8$$

The residual is calculated as the difference between the observed distance and the predicted distance. Given the observed distance is $265$ yards, we compute:

$$\text{Residual} = \text{Observed Distance} - \text{Predicted Distance} = 265 - 260.8333 = 4.199999999999989$$

Rounding to four significant digits, we have:

$$\text{Residual} \approx 4.2$$

Final Answer