Task 1: Modeling Precipitation

Suppose that the precipitation in Chicago can be modeled by a trigonometric function.

Represent time in months elapsed since the beginning of the year (in other words, in January ; February ). The average monthly precipitation for the year is inches, and February is the driest month of the year with inches of precipitation.

1. Identify the independent and dependent variables, both with letter names ( and ) and what they represent in this scenario.

Independent variable: y represents the number of months elapsed since the beginning of the year(time)

Dependent variable: x represents the amount of precipitation in inches(precipitation)

1. Find the amplitude and period of the function. Show your work.

Amplitude: 3.5-2.25=1.25

Period: 12 months(1 year)

1. Write the trigonometric function that represents the expected precipitation for any given month. Explain why you chose your function type, and show work for any values not already given in part B above. y=sin(wx+a)+k a=1.25 k=3.5 Equation: y=1.25sin(/6 x+ x/3)+3.5

1. Graph the function you wrote in part C with technology, such as https://www.desmos.com/calculator and insert the graph below.

(Be sure to choose appropriate window settings.)

1. Predict when the wettest month will be and give the expected precipitation for this month.

Show/explain how you arrived at your conclusions.

Task 2: Modeling Tides

Suppose that the sea level of an inlet is regularly measured at the same point on a bridge and that high and low tides occur in equally spaced intervals. The high tide is observed to be feet above the average sea level of feet; after hours pass, the low tide occurs at feet below the average sea level.

In this task, you will model this occurrence using a trigonometric function by using as a measurement of time. The first high tide occurs at .

1. Identify the independent and dependent variables, both with letter names ( and ) and what they represent in this scenario.

Independent variable: "time" (in hours) represented by x (6 hours)

Dependent variable: measurement (in feet) above or below the sea level, represented by y (10 and 5)

1. Determine these key features of the function that models the tide (show/explain how you found your values for each):
2. Amplitude: 10/2=5
3. Period: 15+5/2=10
4. Frequency: 1/10 cycles per hour
5. Midline:
6. Vertical Shift:
7. Phase Shift:

1. Create a trigonometric function that models the ocean tide.

Explain why you chose your function type. Show work for any values not already outlined above.

y=

1. Graph the function you wrote in part C with technology, such as https://www.desmos.com/calculator and insert the graph below. (Be sure to choose appropriate window settings.)

1. What is the height of the tide at ? Show/explain your work.