The temperature in Fairbanks is approximated by
Upper T left parenthesis x right parenthesis equals 37 sine left bracket StartFraction 2 pi Over 365 EndFraction left parenthesis x minus 101 right parenthesis right bracket plus 25
where​ T(x) is the temperature on day​ x, with xequals1 corresponding to Jan. 1 and xequals365 corresponding to Dec. 31. Estimate the temperature on day 333.
Question content area bottom
Part 1
A.
minus25degrees
B.negative 28 degrees
negative 28degrees
C.330 degrees
330degrees
D.negative 3 degrees
negative 3degrees

The given function for temperature \( T(x) \) is:

\[
T(x) = 37 \sin \left( \frac{2\pi}{365}(x - 101) \right) + 25
\]

We need to estimate the temperature on day \( x = 333 \).

### Step-by-step Calculation:

1. **Substitute \( x = 333 \) into the equation:**

\[
T(333) = 37 \sin \left( \frac{2\pi}{365}(333 - 101) \right) + 25
\]

\[
T(333) = 37 \sin \left( \frac{2\pi}{365}(232) \right) + 25
\]

2. **Simplify the sine argument:**

\[
\frac{2\pi}{365} \times 232 = \frac{464\pi}{365} \approx 3.996\ \text{radians}
\]

3. **Evaluate the sine:**

\[
\sin(3.996) \approx -0.756
\]

4. **Calculate the temperature:**

\[
T(333) = 37 \times (-0.756) + 25
\]

\[
T(333) \approx -27.97 + 25 = -2.97
\]

Thus, the temperature on day 333 is approximately **-3 degrees**.

### The correct answer is:
**D. -3 degrees**

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Would you like a more detailed explanation or further assistance? Here are some related questions for deeper exploration:

1. How is the sine function's behavior relevant to periodic temperature modeling?
2. What is the significance of the constant 101 in this temperature function?
3. How would the temperature differ if you calculated it for day 365?
4. How does the amplitude of 37 affect the temperature extremes?
5. What is the period of this sine function, and how does it relate to the calendar year?

**Tip:** Always check the units of sine arguments when solving periodic functions to ensure correct evaluation.

The temperature in Fairbanks is approximated by T(x) = 37 sin[(2π/365)(x - 101)] + 25, where T(x) is the temperature on day x, with x = 1 corresponding to Jan. 1 and x = 365 corresponding to Dec. 31. Estimate the temperature on day 333.

Learn how to estimate the temperature on day 333 in Fairbanks using the periodic sine function T(x) = 37 sin[(2π/365)(x - 101)] + 25. This trigonometry-based problem involves evaluating sine at specific values and understanding how periodic functions model real-world phenomena. Suitable for high school students in Grades 10-12.

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