Data were collected on the distance a baseball will travel when hit by a baseball bat at a certain speed. The speed, s, is measured in miles per hour, and distance, y, is measured in yards. The line of fit is given by [tex]\hat{y}=3.98+53.59 s[/tex].
If the ball travels for a duration of 5 seconds, what is the predicted distance of the ball?
A. 267.95 yards
B. 271.93 yards
C. 283.87 yards
D. 287.85 yards
Answer (2)
Substitute the given speed, s = 5 miles per hour, into the equation y ^ = 3.98 + 53.59 s .
Calculate the predicted distance: y ^ = 3.98 + 53.59 ( 5 ) = 3.98 + 267.95 = 271.93 yards.
The predicted distance of the ball is 271.93 yards.
Explanation
Understanding the Problem We are given the equation y ^ = 3.98 + 53.59 s , where y ^ is the predicted distance in yards and s is the speed in miles per hour. We are asked to find the predicted distance if the ball travels for a duration of 5 seconds. However, the equation relates distance to speed, not time. It seems there's an error in the problem statement, and the '5 seconds' should actually refer to the speed in miles per hour. Assuming the speed s is 5 miles per hour, we can substitute s = 5 into the equation to find the predicted distance.
Calculating the Predicted Distance Substitute s = 5 into the equation y ^ = 3.98 + 53.59 s :
y ^ = 3.98 + 53.59 ( 5 ) y ^ = 3.98 + 267.95 y ^ = 271.93 Therefore, the predicted distance is 271.93 yards.
Final Answer The predicted distance of the ball is 271.93 yards.
Examples
Understanding linear relationships, like the one in this problem, is useful in many real-world scenarios. For example, if you're planning a road trip, you can use a similar equation to estimate how far you'll travel based on your average speed. Or, if you're tracking the growth of a plant, you can use a linear equation to predict its height based on the number of days since it was planted. These types of equations help us make predictions and understand the relationships between different variables.
The predicted distance of the baseball when hit at a speed of 5 miles per hour is 271.93 yards. I arrived at this answer by substituting the speed into the equation y ^ = 3.98 + 53.59 s . The calculation resulted in the predicted distance being 271.93 yards.
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