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How Fast Can You List Your Favorite Things? If you are asked to make a list of objects in a certain category, how fast you can list them follows a predictable pattern. For example, if you try to name as many vegetables as you can. you'll probably think of several right away— for example, carrots, peas, beans, com. and so on. Then after a pause you might think of ones you eat less frequently—perhaps zucchini, eggplant, and asparagus. Finally, a few more exotic vegetables might come to mind—artichokes, jicama, bok choy, and the like. A psychologist performs this experiment on a number of subjects. The table below gives the average number of vegetables that the subjects named by a given number of seconds.
- (a) Find the cubic polynomial that best fits the data
- (b) Draw a graph of the polynomial from part (a) together with a
scatter plot of the data. - (c) Use your result from part (b) to estimate the number of vegetables that subjects would be able to name in 40 s.
(d) According to the model, how long (to the nearest 0.1 s) would it take a person to name five vegetables?
Seconds | Number of vegetables |
1 | 2 |
2 | 6 |
5 | 10 |
10 | 12 |
15 | 14 |
20 | 15 |
25 | 18 |
30 | 21 |
(a)
![Check Mark](/static/check-mark.png)
To find: The cubic polynomial that best fits the data.
Answer to Problem 3P
The cubic polynomial that best fits the data is
Explanation of Solution
By the use of graphing calculator, the best fit cubic regression
Thus, the quadratic polynomial that best fits the data is
(b)
![Check Mark](/static/check-mark.png)
To draw: The graph of the quadratic polynomial from part (a) with the scatter plot of the data.
Explanation of Solution
Graph:
Consider the seconds as the x coordinates and the number of vegetable a person can name as the y coordinates.
From part (a), the cubic polynomial obtained is
The graph of the cubic polynomial with the scatter plot of the given data is shown below in Figure 1.
From Figure 1, the graph is an continuous and increasing curve.
(c)
![Check Mark](/static/check-mark.png)
To estimate: The number of vegetables a person can able to name in 40s.
Answer to Problem 3P
The number of vegetables a person can able to name in 40s is
Explanation of Solution
From part (b), it is observed that the value of x is 40 occurs when y is 40.9.
Therefore, the number of vegetables a person can able to name in 40s is
(d)
![Check Mark](/static/check-mark.png)
To estimate: The time will it take for a person to name five vegetable.
Answer to Problem 3P
The time will it take for a person to name five vegetable is
Explanation of Solution
From part (a), the model that best fits the data is
From part (b), it is observed that the value of y is 5 occurs when x is 2.01.
Therefore, the time will it take for a person to name five vegetable is
Chapter 3 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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