The Sunrise Question

with solutions…

THE QUESTION [info]: The latest sunrise in Montreal was on Dec. 22 at 9:15 AM . According to the almanac, the earliest sunrise occurred on the 22nd of June at 3:15 AM . The sunrise times on other dates can be predicted using a sinusoidal equation. **Assume there is no daylight savings time in Montreal.**

The latest sunrise in Montreal was on Dec. 22 at 9:15 AM . According to the almanac, the earliest sunrise occurred on the 22nd of June at 3:15 AM . The sunrise times on other dates can be predicted using a sinusoidal equation.

**Assume there is no daylight savings time in Montreal.**

THE QUESTION: Part I (a) Write two equations for the described function above; one using cosine and the other using sine.

(a) Write two equations for the described function above; one using cosine and the other using sine.

THE SOLUTION: Part I STEPS PROCESS Read the given info and convert it into information that can be used in an equation. For example, the time is not going to be written 9:15 on the graph but 9.25 because 15 min. is a quarter of an hour. Make two lists for the parameters A, B, C & D. One set will be used for the cosine equation and the other will be used for the sine equation. Find the parameters in the mean of DABC [stretches before translations]. To find parameter D, add the min. and max. value and then divide by 2 to find the sinusoidal axis . To find parameter A, subtract the average value from the maximum value to get A. cont‘d on the next slide; Hupsha, hupsha now…

STEPS

PROCESS

Read the given info and convert it into information that can be used in an equation. For example, the time is not going to be written 9:15 on the graph but 9.25 because 15 min. is a quarter of an hour.

Make two lists for the parameters A, B, C & D. One set will be used for the cosine equation and the other will be used for the sine equation.

Find the parameters in the mean of DABC [stretches before translations]. To find parameter D, add the min. and max. value and then divide by 2 to find the sinusoidal axis .

To find parameter A, subtract the average value from the maximum value to get A.

STEPS PROCESS cont’d... Parameter B is equal to 2 π divided by the period, which happens to be the number of days in a year; 364. The phase shift (C) is found depending on what kind of equation is being used. If the cosine equation is being found, the maximum value is usually on the y-axis. But the information tells us that the maximum value occurs on Dec. 22, 9 days before Jan. 1 [the y-axis]. The phase shift in the sine equation is determined by finding out the distance of the average value to the y-axis. Finally, to get the equations, plug in the values found into the general formula.

STEPS

PROCESS

cont’d...

Parameter B is equal to 2 π divided by the period, which happens to be the number of days in a year; 364.

The phase shift (C) is found depending on what kind of equation is being used. If the cosine equation is being found, the maximum value is usually on the y-axis. But the information tells us that the maximum value occurs on Dec. 22, 9 days before Jan. 1 [the y-axis].

The phase shift in the sine equation is determined by finding out the distance of the average value to the y-axis.

Finally, to get the equations, plug in the values found into the general formula.

THE QUESTION: Part II (b) Sketch the graph for the sinusoidal function described in the problem.

(b) Sketch the graph for the sinusoidal function described in the problem.

THE SOLUTION: Part II STEPS Acknowledge that Dec. 22 is 9 days before Jan. 1. [use this later in graphing stage.] To find the values within the period of one cycle in the graph, subtract nine days from the full, half, and quarter periods. On the y-axis, label where the min. value, max. value and sinusoidal axis are. Plot the points according to info; max value at Dec. 22. Label the axes.

STEPS

Acknowledge that Dec. 22 is 9 days before Jan. 1. [use this later in graphing stage.]

To find the values within the period of one cycle in the graph, subtract nine days from the full, half, and quarter periods.

On the y-axis, label where the min. value, max. value and sinusoidal axis are. Plot the points according to info; max value at Dec. 22.

Label the axes.

THE GRAPH…

THE QUESTION: Part III (c) Use one of the equations in (a) to predict the time of sunrise on September 7.

(c) Use one of the equations in (a) to predict the time of sunrise on September 7.

THE SOLUTION: Part III Find out what the day of the year September 7 th is by adding up the total number of days in each month up to the given date, assuming there is no daylight savings time. Since d lies along the x-axis, treat the number of days as an x-coordinate and plug in as d in the formula, either sine or cosine and solve. Sun, sun...Mr. Golden Sun.

Find out what the day of the year September 7 th is by adding up the total number of days in each month up to the given date, assuming there is no daylight savings time.

Since d lies along the x-axis, treat the number of days as an x-coordinate and plug in as d in the formula, either sine or cosine and solve.

Sun, sun...Mr. Golden Sun.

THE QUESTION: Part IV (d) What is the average sunrise time throughout the year?

(d) What is the average sunrise time throughout the year?

THE SOLUTION: Part IV STEPS PROCESS The avg. sunrise time equals the sinusoidal axis. To find this, find parameter D by adding the minimum and maximum value and divide by two.

STEPS

PROCESS

The avg. sunrise time equals the sinusoidal axis. To find this, find parameter D by adding the minimum and maximum value and divide by two.

Looking at the sun gives me a feeling of freedom. Too bad Max can’t feel this way… Sunrise Over Rocks, Lighthouse Beach by flickr user Captain Capture

Sunrise Over Rocks, Lighthouse Beach by flickr user Captain Capture