Concept explainers
Hours of Daylight According to the Old Farmer's Almanac, in Anchorage, Alaska, the number of hours of sunlight on the summer solstice of 2015 was , and the number of hours of sunlight on the winter solstice was .
(a) Find a sinusoidal function of the form that models the data.
(b) Use the function found in part to predict the number of hours of sunlight on April 1, the 91st day of the year.
(c) Draw a graph of the function found in part .
(d) Look up the number of hours of sunlight for April 1 in the Old Farmer's Almanac, and compare the actual hours of daylight to the results found in part .
To find:
a. Find a sinusoidal function of the form that models the data.
Answer to Problem 37AYU
Solution:
a.
Explanation of Solution
Given:
According to the Old Farmer’s Almanac, in Anchorage, Alaska, the number of hours of sunlight on the summer solstice of 2015 was , and the number of hours of sunlight on the winter solstice was .
Calculation:
a. Find a sinusoidal function of the form that models the data.
Determine A, the amplitude of the function.
Determine B, the vertical shift of the function.
Determine , It is easier to find the horizontal stretch factor first. Since the period of the function is . Because . Therefore .
Now superimpose the graph of
Determine the horizontal shift of the function by using the period of the data.
Since sine curve is increasing in the interval and decreasing in the interval . So a local maximum point occurs at . Since the maximum occurs at . So we must shift the graph of the function units to the right by replacing by
To find:
b. Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year.
Answer to Problem 37AYU
Solution:
b. hours
Explanation of Solution
Given:
According to the Old Farmer’s Almanac, in Anchorage, Alaska, the number of hours of sunlight on the summer solstice of 2015 was , and the number of hours of sunlight on the winter solstice was .
Calculation:
b. Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year.
So we predict that there will be about
To find:
c. Draw a graph of the function found in part (a).
Answer to Problem 37AYU
Solution:
c.
Explanation of Solution
Given:
According to the Old Farmer’s Almanac, in Anchorage, Alaska, the number of hours of sunlight on the summer solstice of 2015 was , and the number of hours of sunlight on the winter solstice was .
Calculation:
c. Draw a graph of the function found in part (a).
To find:
d. Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b).
Answer to Problem 37AYU
Solution:
d. The actual hours of sunlight on April 1, 2015, were hours. This is close to the predicted amount of hours.
Explanation of Solution
Given:
According to the Old Farmer’s Almanac, in Anchorage, Alaska, the number of hours of sunlight on the summer solstice of 2015 was , and the number of hours of sunlight on the winter solstice was .
Calculation:
d. Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac, and compare the actual hours of daylight to the results found in part (b).
The actual hours of sunlight on April 1, 2015, were hours. This is close to the predicted amount of hours.
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