At Whitehorse in the Yukon, the following facts are available from the Internet.
On June 21 ﻿the sun rises at 03:20 (Standard Time)
On December 21 ﻿the sun rises at 10:14 (Standard Time)
The function describing the sunrise times is sinusoidal with a period of 365 ﻿days.
a. ﻿Determine an equation for this function.
b. ﻿Determine the time of the sunrise for July 1.

To solve this problem, we need to model the sunrise times as a sinusoidal function with a period of 365 days.

### Part a: Determine the equation for the sunrise function

We are given that:

- The sunrise time is modeled by a sinusoidal function.
- The period is \(365\) days.
- On **June 21**, the sunrise occurs at 03:20 AM (which we'll express as 3.33 hours in decimal form).
- On **December 21**, the sunrise occurs at 10:14 AM (which we'll express as 10.23 hours in decimal form).

#### Step 1: Model the sinusoidal function
A general sinusoidal function can be written as:

\[
y(t) = A \sin\left(B(t - C)\right) + D
\]

Where:
- \(A\) is the amplitude (half the difference between the maximum and minimum values),
- \(B\) is related to the period,
- \(C\) is the horizontal shift,
- \(D\) is the vertical shift.

#### Step 2: Determine \(B\)
The period is \(365\) days, so:

\[
B = \frac{2\pi}{365}
\]

#### Step 3: Determine \(D\) (the vertical shift)
The vertical shift \(D\) represents the average sunrise time, which is the midpoint between the earliest and latest sunrise times.

- Earliest sunrise: June 21 at 03:20 AM \(\approx 3.33\) hours.
- Latest sunrise: December 21 at 10:14 AM \(\approx 10.23\) hours.

So, the average (midpoint) sunrise time is:

\[
D = \frac{3.33 + 10.23}{2} = 6.78 \text{ hours} \quad (\text{about 6:47 AM})
\]

#### Step 4: Determine \(A\) (the amplitude)
The amplitude \(A\) is half the difference between the maximum and minimum sunrise times:

\[
A = \frac{10.23 - 3.33}{2} = 3.45 \text{ hours}
\]

#### Step 5: Determine \(C\) (the horizontal shift)
Since June 21 corresponds to the earliest sunrise (minimum value), we can shift the function accordingly. June 21 is the 172nd day of the year, so \(C = 172\).

### Equation
Now we can substitute these values into the general sinusoidal function:

\[
y(t) = 3.45 \sin\left(\frac{2\pi}{365}(t - 172)\right) + 6.78
\]

Where \(y(t)\) is the sunrise time in hours, and \(t\) is the day of the year (with \(t = 1\) corresponding to January 1st).

### Part b: Determine the sunrise time for July 1

July 1 is the 182nd day of the year. We substitute \(t = 182\) into the equation to find the sunrise time:

\[
y(182) = 3.45 \sin\left(\frac{2\pi}{365}(182 - 172)\right) + 6.78
\]

Let's calculate this.The sunrise time on July 1st is approximately 7.37 hours, which is around 7:22 AM.

### Summary of the results:
- The equation describing the sunrise times is:
  \[
  y(t) = 3.45 \sin\left(\frac{2\pi}{365}(t - 172)\right) + 6.78
  \]
- The sunrise time on July 1st is about **7:22 AM**.

Do you have any questions or would you like further details on this solution?

Here are 5 follow-up questions:
1. How would the equation change if the period were different?
2. How would you determine the sunset times using a similar approach?
3. What is the significance of the amplitude in this sinusoidal function?
4. How would daylight saving time affect this model?
5. Could we use a similar sinusoidal model to predict temperature variations?

**Tip:** In sinusoidal functions, the amplitude represents half the range of the values, showing how much the function deviates from its average.

At Whitehorse in the Yukon, the following facts are available from the Internet. On June 21 the sun rises at 03:20 (Standard Time). On December 21 the sun rises at 10:14 (Standard Time). The function describing the sunrise times is sinusoidal with a period of 365 days. a. Determine an equation for this function. b. Determine the time of the sunrise for July 1.

Explore how to model sunrise times in Whitehorse, Yukon, with a sinusoidal function. Given specific sunrise times for June 21 and December 21, this problem provides a step-by-step guide to finding the equation and determining the sunrise time for July 1. Suitable for advanced high school students.

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