Create a roller coaster function R(t) in DESMOS, where t is measured in seconds and R(t), in feet, measures the height of the roller coaster at time t. Write the equation using appropriate notation and name your roller coaster. R(t) must meet the minimum requirements: A. The segment of your coaster ride must have at least 3 relative maxima and/or minima. B. The ride dives below the ground into a tunnel at least once. C. State the x values that you are using to illustrate your part of the roller coaster. 2. Describe where your polynomial is increasing and decreasing. 3. What is the highest point your roller coaster goes? 4. What is the domain that represents your segment of the coaster? 5. Which interval is the ‘scariest’ and why? 6. What is the degree of your polynomial? 7. What is the leading coefficient? 8. How many real zeros does your polynomial have? What do these mean in the context of the problem?
The Correct Answer and Explanation is:
To create a roller coaster function R(t)R(t)R(t), we can use a polynomial that fits all the criteria. Here’s a possible equation for the roller coaster:R(t)=−0.1(t−2)(t−4)(t−7)(t−10)R(t) = -0.1(t – 2)(t – 4)(t – 7)(t – 10)R(t)=−0.1(t−2)(t−4)(t−7)(t−10)
Where:
- ttt is the time in seconds,
- R(t)R(t)R(t) is the height of the roller coaster in feet.
This is a polynomial of degree 4, as it has four factors. Now let’s walk through the answers to each part of the problem:
1. Equation and Description of the Roller Coaster
The roller coaster function is:R(t)=−0.1(t−2)(t−4)(t−7)(t−10)R(t) = -0.1(t – 2)(t – 4)(t – 7)(t – 10)R(t)=−0.1(t−2)(t−4)(t−7)(t−10)
- The coaster starts at t=0t = 0t=0, peaks at around 20 feet at t=2t = 2t=2, dips to a minimum around t=5t = 5t=5, then rises again around t=7t = 7t=7, and dives below ground (through a tunnel) near t=10t = 10t=10.
2. Increasing and Decreasing Segments
- The polynomial increases between t=0t = 0t=0 and t=2t = 2t=2, then decreases between t=2t = 2t=2 and t=5t = 5t=5, increases again between t=5t = 5t=5 and t=7t = 7t=7, and finally decreases between t=7t = 7t=7 and t=10t = 10t=10.
3. Highest Point
- The highest point of the roller coaster occurs at the local maxima at t=2t = 2t=2, where the height R(t)R(t)R(t) reaches around 20 feet.
4. Domain of the Roller Coaster
- The domain of the roller coaster is the range of time values during which the ride happens. This is given by t∈[0,10]t \in [0, 10]t∈[0,10] because the coaster’s path starts at t=0t = 0t=0 and ends at t=10t = 10t=10.
5. Scariest Interval
- The “scariest” interval could be the one where the coaster dips below the ground, which happens between t=7t = 7t=7 and t=10t = 10t=10. The tunnel effect and the drop create a thrilling, intense feeling, making it the most exciting or ‘scary’ part of the ride.
6. Degree of the Polynomial
- The degree of the polynomial is 4 because there are four factors in the equation: (t−2)(t−4)(t−7)(t−10)(t – 2)(t – 4)(t – 7)(t – 10)(t−2)(t−4)(t−7)(t−10).
7. Leading Coefficient
- The leading coefficient of this polynomial is −0.1-0.1−0.1. This is the coefficient of the highest-degree term, which is t4t^4t4.
8. Real Zeros of the Polynomial
- The polynomial has four real zeros: t=2,4,7,10t = 2, 4, 7, 10t=2,4,7,10. These zeros represent the points where the roller coaster reaches the ground level (height R(t)=0R(t) = 0R(t)=0). At t=2,4,7,10t = 2, 4, 7, 10t=2,4,7,10, the roller coaster either starts, dips below, or finishes the ride.
In summary, the roller coaster is a degree 4 polynomial, with a domain from t=0t = 0t=0 to t=10t = 10t=10. The ride has multiple increases and decreases, with several relative maxima and minima, and it dives below the ground at one point, meeting all the necessary requirements for the ride.
