Concept explainers
Height of a Baseball A baseball is thrown upward, and its height is measured at 0.5-s intervals using a strobe light. The resulting data are given in the table.
- (a) Draw a scatter plot of the data. What degree polynomial is appropriate for modeling the data ?
- (b) Find a polynomial model that best fits the data, and graph it on the scatter plot.
- (c) Find the times when the ball is 20 ft above the ground.
(d) What is the maximum height attained by the ball?
Time (s) | Height (ft) |
0 | 4.2 |
0.5 | 26.1 |
1.0 | 40.1 |
1.5 | 46.0 |
2.0 | 43.9 |
2.5 | 33.7 |
3.0 | 15.8 |
(a)
To draw: The scatter plot of the data and to find appropriate degree of the polynomial for modeling the data.
Explanation of Solution
Consider the time as the x coordinates and the height of the baseball from the ground as the y coordinates.
From Figure 1, the given data are plotted on the graph. The ball reaches maximum height at (1.5, 46).
It is appropriate to use quadratic polynomial to model the given data, if there is single peak in the given data.
It is observed that the given data appears to have a peak. Therefore, it is appropriate to use quadratic polynomial (degree 2) as a model for the given data.
(b)
To find: The polynomial model that best fits the data and graph it with the scatter plot.
Answer to Problem 5P
The quadratic polynomial that best fits the data is
Explanation of Solution
From part (a), it is appropriate to use quadratic polynomial (degree 2) as a model for the given data.
By the use of graphing calculator, the best fit quadratic regression
Thus, the quadratic polynomial that best fits the data is
The graph of the quadratic polynomial with the scatter plot of the given data is shown below in Figure 2.
From Figure 2, the graph is a downward parabola and the given data are plotted on the graph.
(c)
To find: The time when the ball is at height of 20 ft above the ground.
Answer to Problem 5P
The time when the ball is at height of 20 ft above the ground is
Explanation of Solution
From part (b), it is observed that the value of y is 20 occurs when x is 0.34.
Therefore, the time when the ball is at height of 20 ft above the ground is
(d)
To find: The maximum height attained by the ball.
Answer to Problem 5P
The maximum height attained by the ball is
Explanation of Solution
From Figure 2, it is observed that the maximum point of the parabola occurs at y = 46.20.
Therefore, maximum height attained by the ball is
Chapter 3 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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