Final answer:
To verify factors of the polynomial w(x), we use synthetic division for x=2, x=5, and x=8, and direct substitution for x=-4 and x=-1. The water slide's starting height is 3.2 feet, and the duration at or below ground level is found by setting w(x) ≤ 0. The highest point is found through calculus or graph analysis, and the polynomial has a degree of 5 with a leading coefficient of -0.01, influencing its end behavior.
Step-by-step explanation:
To confirm the factors of x=2, x=5, and x=8 for the function w(x) = -0.01x5 + 0.1x4 + 0.05x3 - 1.9x2 + 1.36x + 3.2, we use synthetic division for each value of x and ensure the remainder is 0. When direct substitution is used for x=-4 and x=-1, the resulting value of w(x) should be 0 if they are indeed factors.
1) The height off the ground of the platform to start the water slide is given by w(0), which is 3.2 feet. We know this because when x=0, it indicates the start.
2) The water slide is at or below ground level where w(x) ≤ 0. We find the durations where this occurs by determining the x-values (time in seconds) where the water slide's height is at or below 0 and add up those intervals.
3) To find the highest point the water slide reaches, we must either calculate the maximum value of w(x) using calculus (finding the critical points and evaluating the function at these points) or by analyzing the graph of the function.
4) The degree of w(x) is 5, as the highest exponent of x is 5. The leading coefficient is -0.01, which affects the end behavior of the polynomial; as x approaches infinity, w(x) approaches negative infinity, and as x approaches negative infinity, w(x) also approaches negative infinity.