Math  /  Trigonometry

QuestionFor the given function, use your intuition or additional research, if necessary, to complete parts (a) through (c) below. (angle of football, horizontal distance traveled by football) a. Describe an appropriate domain and range for the function. b. Make a rough sketch of a graph of the function. c. Briefly discuss the validity of the graph as a model of the true function. a. Choose the appropriate domain for the function below. A. The domain is the horizontal distances traveled by the football thrown at certain angles or 0 ft to 200 mi . B. The domain is all angles a football could be thrown at or 00^{\circ} to 360360^{\circ}. C. The domain is all angles a football could be thrown at or 00^{\circ} to 9090^{\circ}. D. The domain is the horizontal distances traveled by the football thrown at certain angles or 0 ft to 200 ft .

Studdy Solution

STEP 1

1. The function relates the angle at which a football is thrown to the horizontal distance it travels.
2. We need to determine the appropriate domain and range, sketch the graph, and discuss the model's validity.

STEP 2

1. Determine the appropriate domain for the function.
2. Determine the appropriate range for the function.
3. Sketch a rough graph of the function.
4. Discuss the validity of the graph as a model.

STEP 3

The domain of the function should be the set of all possible angles at which a football can be thrown.
Considering practical scenarios, the angle should be between 00^\circ and 9090^\circ because angles beyond 9090^\circ would result in the football being thrown backward or vertically.
Thus, the appropriate domain is: C. The domain is all angles a football could be thrown at or 0 to 90. \text{C. The domain is all angles a football could be thrown at or } 0^\circ \text{ to } 90^\circ.

STEP 4

The range of the function should be the set of all possible horizontal distances the football can travel when thrown at angles within the domain.
The horizontal distance will vary from 00 feet (at 00^\circ or 9090^\circ) to some maximum distance achieved at an optimal angle.
Thus, the range is from 00 feet to the maximum distance achieved.

STEP 5

Sketch a rough graph of the function.
The graph should start at 00 feet at 00^\circ, rise to a maximum distance at an optimal angle (usually around 4545^\circ), and return to 00 feet at 9090^\circ.
The graph will resemble a concave down parabola.

STEP 6

Discuss the validity of the graph as a model.
The graph is a simplified model of the true function. It assumes no air resistance and a constant initial speed. In reality, factors like air resistance, wind, and varying initial speeds can affect the actual trajectory and distance.
The model is valid for basic predictions and understanding the relationship between angle and distance but may not be accurate for precise calculations.

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