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Can you imagine analyzing the elk population problem without technology? It can be done, but it would take a long time to work out the maximum point and the intercepts by hand. In fact, finding the exact locations of the extrema by hand requires knowledge of calculus!
Let’s explore more modeling situations that require the use of technology. Read through the problem on each tab below. Answer the questions, then check your answers.
Between the years of 1989 and 1997, the percent of households with incomes of $100,000 or more were recorded. The data are reported in the table below.
| Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Percent | 8.05 | 7.695 | 7.482 | 7.411 | 7.482 | 7.695 | 8.05 | 8.547 | 9.186 |
Find a polynomial function that models these data.
In what year did the percent of families whose incomes were more than $100,000 reach its minimum?
Complete each step, then click it to check your answer.
The finite differences method indicates a quadratic model.
The polynomial equation that models these data is \( y = 0.071x^2 – 0.426x + 8.05 \).
Rewrite this equation using the variables P and y, where P is the percent of households and y is the year.
\( P(y) = 0.071y^2 – 0.426y + 8.05 \)
Add the curve to the points on the coordinate plane.
The minimum value is (3, 7.411), which means that at year three, a minimum number of households had an income of $100,000 or more.
Since the data start in 1989 (year 0), the minimum was reached in 1992.
The start of a roller coaster hill is about 2 meters off the ground. The height of the roller coaster train, as it climbs the hill, is recorded at each half-second in the table below.
| Time (sec) | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| Height (m) | 2 | 6.25 | 16 | 26.75 | 34 | 33.25 | 20 |
What polynomial function models this roller coaster hill?
What is its maximum height?
When does the roller coaster train return to its starting height?
Complete each step, then click it to check your answer.
Based on experience, it looks like it could be cubic. You will need to use the finite differences method to make sure.
The finite differences method shows us that the model will be a cubic model.
The polynomial equation that models this data is \( y = –6x^3 + 20x^2 + 2 \).
Rewriting with the variables H and t, \( H(t) = –6t^3 + 20t^2 + 2 \), where H is the height of the roller coaster train and t is the time.
Graph the polynomial function so that you can see both the maximum height and where it returns to 2 meters on the y-axis.
The relative maximum of the polynomial model is (2.222, 34.922), which means that the roller coaster would peak at 2.222 seconds at a height of almost 35 meters.
The polynomial model also shows that (3.333, 2.022) is a point on the line. This would indicate that the roller coaster train returns to its starting height at about 3.33 seconds after entering the hill.
