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 degrees40°

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 bracketD(t)=12−3cos2π(t+10)365​,

where D is measured in hours and

0less than or equals≤tless than or equals≤365

is measured in​ days, with

tequals=0

corresponding to January 1. Complete parts​ (a) through​ (e) below.

**e.** At what times of year is the length of day changing most​ rapidly? Least​ rapidly?

The length of day is changing most rapidly during days

tequals=

To determine when the length of the day is changing most and least rapidly, we need to analyze the rate of change of the daylight hours function \( D(t) \). This can be done by finding the derivative of the function \( D(t) \) with respect to \( t \), as the derivative represents the rate of change.

### Step 1: Derivative of the function \( D(t) \)
Given the function:
\[
D(t) = 12 - 3 \cos \left( \frac{2\pi (t + 10)}{365} \right)
\]
we can find the derivative \( D'(t) \).

### Step 2: Use the chain rule
First, the derivative of the constant term \( 12 \) is \( 0 \), so we only need to differentiate the second term. The derivative of \( \cos(x) \) is \( -\sin(x) \), and then we apply the chain rule:

\[
D'(t) = 3 \cdot \sin \left( \frac{2\pi (t + 10)}{365} \right) \cdot \frac{d}{dt} \left( \frac{2\pi (t + 10)}{365} \right)
\]

The derivative of \( \frac{2\pi (t + 10)}{365} \) with respect to \( t \) is \( \frac{2\pi}{365} \), so we get:

\[
D'(t) = 3 \cdot \sin \left( \frac{2\pi (t + 10)}{365} \right) \cdot \frac{2\pi}{365}
\]

Thus, the rate of change of daylight hours is:

\[
D'(t) = \frac{6\pi}{365} \sin \left( \frac{2\pi (t + 10)}{365} \right)
\]

### Step 3: Analyze when the rate of change is greatest and least
The rate of change \( D'(t) \) is proportional to the sine function \( \sin \left( \frac{2\pi (t + 10)}{365} \right) \). The sine function oscillates between \( -1 \) and \( 1 \), and it reaches its maximum when \( \sin \left( \frac{2\pi (t + 10)}{365} \right) = 1 \) and minimum when \( \sin \left( \frac{2\pi (t + 10)}{365} \right) = -1 \).

- **Maximum rate of change (most rapidly):** This occurs when \( \sin \left( \frac{2\pi (t + 10)}{365} \right) = \pm 1 \), i.e., the sine function achieves its maximum values. These correspond to the spring equinox and the fall equinox, approximately on **March 21 (t ≈ 80)** and **September 21 (t ≈ 265)**.
  
- **Minimum rate of change (least rapidly):** This occurs when \( \sin \left( \frac{2\pi (t + 10)}{365} \right) = 0 \), i.e., the sine function is at zero. These correspond to the solstices—**winter solstice** around **December 21 (t ≈ 355)** and **summer solstice** around **June 21 (t ≈ 172)**.

### Final Answer:
- The length of the day is changing most rapidly around **March 21** and **September 21** (spring and fall equinox).
- The length of the day is changing least rapidly around **December 21** and **June 21** (winter and summer solstice).

---

Would you like more details on how the derivative was calculated? 

Here are 5 related questions to explore:
1. How do we interpret the value of \( D'(t) \) in terms of the rate of daylight change?
2. What does the cosine function in \( D(t) \) represent physically?
3. How could you approximate the length of the day on any given day using \( D(t) \)?
4. How would the function change if we considered a location at a different latitude?
5. Why does the sine function govern the rate of daylight change over the year?

**Tip:** Always check the behavior of sine and cosine functions over their cycles to identify maximum and minimum rates of change in periodic phenomena.

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° north latitude, the length of a day is approximated by D(t) = 12 - 3 cos(2π(t+10)/365), where D is measured in hours and t is measured in days, with t=0 corresponding to January 1. At what times of year is the length of day changing most rapidly? Least rapidly?

Explore how the rate of change in daylight hours varies throughout the year at 40° north latitude. Learn how calculus and trigonometric functions explain the most rapid and least rapid changes in day length, with key dates like the equinoxes and solstices. Suitable for students in Grades 10-12.

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