Question

in the month of march, the temperature at the south pole varies over the day in a periodic way that can be modeled approximately by a trigonometric function. the highest temperature is about -50 degrees Celsius, and its reached around 2 p.m. the lowest temperature -54 degrees Celsius and it is reached half a day apart from the highest temperature, at 2 a.m. Find the formula of the trigonometric function that models the temperature T in the South Pole in March t hours after midnight. Define the function using radians.

Answer 1

2 am

Step-by-step explanation:

We know that the temperature variation is depicted by trignometric function. The total cycle of any trignometric function can be divded into four parts. In first quarter of the cycle, the value rises from mean to max. In 2nd quater, it falls to mean again. In third quarter it lowers to minimum and in last quarter rises to mean again.Dividing the daily temperature variation cycle into four quaters of 6 hours each. The difference in time for any trignometric function to reach from its maximum value to minimum value is one-half of the cycle. In this case, it will be 12 hours.Therefore daily temperature will fall to its minimum value 12 hours after reaching maximum that is at 2 am

Answer 2

The formula for the trigonometric function that models the temperature T in the South Pole in March, t hours after midnight, using radians, is:

T(t) = -52 - 2 cos(π(t - 6) / 12)

where:

T(t) is the temperature in degrees Celsius at t hours after midnight (0 ≤ t ≤ 24)

-52 is the average temperature between the highest and lowest points (midline)

2 is the amplitude, representing the difference between the average and the extreme temperatures

π is the mathematical constant pi (approximately 3.14159)

t - 6 represents the time shifted 6 hours forward from midnight to account for the peak at 2 pm

12 is the period of the function in hours, representing the daily cycle

Step-by-step explanation:

Modeling the Temperature:

We can model the temperature as a sinusoidal function because it follows a periodic pattern with a maximum and minimum value within a specific period.

Midline, Amplitude, and Period:

The midline is halfway between the highest and lowest temperatures, which is -52°C.

The amplitude is half the difference between the extreme temperatures, which is 2°C.

The period is the time it takes for the temperature to complete one full cycle (24 hours).

Phase Shift:

The highest temperature is reached at 2 pm, which is 6 hours after midnight. To account for this phase shift, we subtract 6 from the time variable (t) within the cosine function.

Putting it Together:

Combining these elements, we get the formula:

T(t) = -52 - 2 cos(π(t - 6) / 12)

This formula gives the temperature in degrees Celsius at any time t hours after midnight in March at the South Pole, based on the given information about the temperature cycle.

Note: This is a simplified model and may not perfectly capture all the nuances of the actual temperature variations. However, it provides a good approximation based on the available information.