The number of hours of daylight at any point on Earth fluctuates throughout the year. In the northern​ hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40 degrees north​ latitude, the length of a day is approximated by Upper D left parenthesis t right parenthesis equals 12 minus 3 cosine left bracket StartFraction 2 pi left parenthesis t plus 10 right parenthesis Over 365 EndFraction right bracket
​, where D is measured in hours and 0less than or equalstless than or equals365 is measured in​ days, with tequals0 corresponding to January 1. Complete parts​ (a) through​ (e) below.
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Part 1
a. Approximately how much daylight is there on October 28​? ​(tequals300​)
The amount of daylight on October 28 is

enter your response here hours.

We are given the formula for the length of a day \( D(t) \) as a function of \( t \), which is the number of days since January 1. The formula is:

\[
D(t) = 12 - 3 \cos \left( \frac{2 \pi (t + 10)}{365} \right)
\]

We are asked to find the number of daylight hours on October 28, where \( t = 300 \).

### Step-by-Step Solution:

1. **Substitute \( t = 300 \) into the equation:**

   \[
   D(300) = 12 - 3 \cos \left( \frac{2 \pi (300 + 10)}{365} \right)
   \]

   Simplifying inside the parentheses:

   \[
   D(300) = 12 - 3 \cos \left( \frac{2 \pi (310)}{365} \right)
   \]

2. **Calculate the angle:**

   \[
   \frac{2 \pi \times 310}{365} = \frac{620 \pi}{365} = \frac{124 \pi}{73}
   \]

3. **Find the cosine of the angle:**
   Now calculate \( \cos \left( \frac{124 \pi}{73} \right) \). This is approximately:

   \[
   \cos \left( \frac{124 \pi}{73} \right) \approx \cos(5.34 \text{ radians})
   \]

4. **Substitute the value of the cosine and compute \( D(300) \):**

Now, I'll calculate the value for the cosine and the final daylight hours.The amount of daylight on October 28 at 40 degrees north latitude is approximately 10.25 hours.

Would you like more details or have any questions about this?

Here are 5 related questions you might find interesting:
1. How does the formula for daylight change if you are at a different latitude?
2. What is the daylight duration on the summer solstice using this formula?
3. How does cosine function influence the periodic nature of daylight throughout the year?
4. Can this formula be used for other celestial events like sunrise or sunset?
5. How does daylight duration compare between different hemispheres?

**Tip:** The cosine function helps model periodic behaviors, such as daylight hours, because it naturally oscillates between maximum and minimum values over time.

Approximately how much daylight is there on October 28 at 40 degrees north latitude, using the formula D(t) = 12 - 3 cos [(2π(t + 10)/365)], where t = 300?

Find out how to calculate the hours of daylight on October 28 at 40 degrees north latitude using the formula D(t) = 12 - 3 cos [(2π(t + 10)/365)]. This trigonometry-based problem illustrates periodic functions, cosine, and the relationship between Earth’s rotation and daylight hours. Suitable for high school students in Grades 10-12.

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