38. MODELING REAL LIFE

The function [tex]f(x) = -200x + 1000[/tex] represents the altitude (in feet) of a paraglider from the time the paraglider begins a descent to a landing site located 100 feet above sea level.

a. Find the domain and range in this context. Then graph the function.

b. Interpret the slope and intercepts of the graph.

c. The function [tex]g(x) = -150x + 900[/tex] represents the altitude (in feet) of a second paraglider x minutes from the time the paraglider begins a descent to the same landing site. Both paragliders start their descent at the same time. Who reaches an altitude of 100 feet first? Explain.

The domain of the function is the set of possible times for the paraglider's descent, while the range is the set of possible altitudes. The graph of the function shows the altitude decreasing linearly with time. The slope represents the rate of descent, and the intercept represents the starting altitude.

Explanation:

The domain of a function represents the possible input values for that function. In this context, since the function represents the altitude of a paraglider during descent, the domain would be the set of all possible times during the descent. In this case, the time would start when the paraglider begins the descent and continue until it lands. The range of a function represents the possible output values for that function. In this case, since the function represents altitude, the range would be all the possible altitudes during the descent, which could go from sea level (0 feet) to the altitude of the landing site (100 feet above sea level).

To graph the function, you can choose different values for x (time) and then calculate the corresponding values for f(x) (altitude). A simple way to graph this linear function is to plot two points and draw a straight line through them. For example, if you choose x = 0, f(x) would be 1000 (since the paraglider starts at an altitude of 1000 feet) and if you choose x = 5, f(x) would be 600 (since the paraglider descends 200 feet every minute). Connecting these two points with a straight line will give you the graph of the function.

The slope of the graph represents the rate of change of altitude with respect to time. In this case, the slope is -200, which means that for every minute that passes, the altitude decreases by 200 feet. The intercept of the graph represents the initial altitude when the paraglider begins its descent. In this case, the intercept is 1000, indicating that the paraglider starts at an altitude of 1000 feet.

To determine who reaches an altitude of 100 feet first, you can set up an equation with the given altitudes and solve for x. For the first paraglider, f(x) = 100 (since it reaches an altitude of 100 feet), and for the second paraglider, g(x) = 100. Solving these equations will give you the respective times when each paraglider reaches an altitude of 100 feet. You can then compare the two times to determine which paraglider reaches the altitude first.

Learn more about Domain and Range of Functions here:

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