Question

The data in the table represents the average number of daylight hours each month in Springfield in 2015, rounded to the nearest quarter-hour.

Month Hours
January 9.5
February 10.5
March 12
April 13.25
May 14.5
June 15
July 14.75
August 13.75
September 12.5
October 11.0
November 9.75
December 9.25

Write an equation that best models the data.

What is the expected number of daylight hours in March 2020? Explain.
(4 points)

Skip to navigation

Answer 1

To create an equation that models the average number of daylight hours per month in Springfield from the provided data, we can treat the months of the year as inputs and the corresponding hours as outputs. We can assign numerical values to each month, where January is 1, February is 2, March is 3, and so on, up to December which is 12.

Here are the data points in (x, y) form, where \( x \) is the month number and \( y \) is the number of daylight hours:

1. (1, 9.5)
2. (2, 10.5)
3. (3, 12)
4. (4, 13.25)
5. (5, 14.5)
6. (6, 15)
7. (7, 14.75)
8. (8, 13.75)
9. (9, 12.5)
10. (10, 11.0)
11. (11, 9.75)
12. (12, 9.25)

### Finding an Equation

Given the data is periodic with peaks in June and July, a sinusoidal function could be appropriate. A general form of a cosine function can be:

\[
y = A \cos(B(x - C)) + D
\]

Where:
- \( A \) is the amplitude (half of the difference between max and min values),
- \( B \) relates to the period of the function (full cycle over the year),
- \( C \) is the horizontal shift (phasing),
- \( D \) is the vertical shift (baseline level).

1. **Calculate the amplitude (A):**

Maximum = 15 (June)

Minimum = 9.25 (December)

Amplitude \( A \):
\[
A = \frac{15 - 9.25}{2} = \frac{5.75}{2} = 2.875
\]

2. **Calculate the vertical shift (D):**

\[
D = \frac{15 + 9.25}{2} = \frac{24.25}{2} = 12.125
\]

3. **Find B:**

The period of daylight changes is 12 months, so:
\[
B = \frac{2\pi}{12} = \frac{\pi}{6}
\]

4. **Determine C:**

The maximum occurs at \( x = 6 \) (June), so we would want the cosine function to be at its maximum at this point. Thus:
\[
C = 6
\]

The general equation is:
\[
y = 2.875 \cos\left(\frac{\pi}{6}(x - 6)\right) + 12.125
\]

### Finding the Expected Daylight Hours in March 2020

To find the expected number of daylight hours in March (which corresponds to \( x = 3 \)), we substitute \( x = 3 \) into the equation:

\[
y = 2.875 \cos\left(\frac{\pi}{6}(3 - 6)\right) + 12.125
\]
\[
= 2.875 \cos\left(\frac{\pi}{6} \cdot (-3)\right) + 12.125
\]
\[
= 2.875 \cos\left(-\frac{3\pi}{6}\right) + 12.125
\]
\[
= 2.875 \cos\left(-\frac{\pi}{2}\right) + 12.125
\]
\[
= 2.875 \cdot 0 + 12.125
\]
\[
= 12.125
\]

### Conclusion

Therefore, the expected number of daylight hours in March 2020 is approximately **12.13 hours**. This method assumes the same daylight hour patterns will continue consistently as observed in 2015.