The number of hours of sunlight in a particular location S(t) can be modeled by the function S(t)=12+2cos(2pi/365t), where t is the number of days after January 1st. (That is, t=0 means January 1st.) After how many days will there be 12 hours of sunlight for the first time during the year?

The function modeling the number of hours of sunlight is given by:

\[
S(t) = 12 + 2 \cos\left(\frac{2\pi}{365}t\right)
\]

We need to find the first time \( t \) when there are 12 hours of sunlight. This occurs when \( S(t) = 12 \).

Set the equation equal to 12 and solve for \( t \):

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

Subtract 12 from both sides:

\[
2 \cos\left(\frac{2\pi}{365}t\right) = 0
\]

Divide both sides by 2:

\[
\cos\left(\frac{2\pi}{365}t\right) = 0
\]

The cosine function equals zero at:

\[
\frac{2\pi}{365}t = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots
\]

We only need the first instance where this occurs, which is when:

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

Solve for \( t \):

\[
t = \frac{\pi/2 \times 365}{2\pi} = \frac{365}{4} = 91.25
\]

Since \( t \) represents the number of days after January 1st, we round to the nearest whole number. Therefore, the first time there will be 12 hours of sunlight is on day \( t = 91 \).

Thus, **there will be 12 hours of sunlight for the first time on the 91st day of the year, which is April 1st**.

Would you like any details or have any further questions?

Here are some related questions you might find interesting:
1. How would the function change if the maximum number of sunlight hours was 14 instead of 12?
2. What would the model look like if the minimum number of sunlight hours were 8?
3. How does the period of the cosine function relate to the length of the year?
4. What are the implications of the cosine function's amplitude in this context?
5. How would you determine the day of the year with the maximum number of sunlight hours?

**Tip:** The cosine function is commonly used in modeling periodic phenomena like sunlight hours because of its smooth, wave-like behavior.

Learn how to find the first day with 12 hours of sunlight using the function S(t) = 12 + 2 cos(2π/365 t). Understand the properties of the cosine function and its application in modeling sunlight hours throughout the year. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation