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 ° C and it is reached around 2 p.m. The lowest temperature is about −54 ° C 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. What is the temperature at 5 p.m? Round your answer if necessary to two decimal places.

A. ( T(t) = -2cosleft(π/12tright) - 52 )
B. ( T(t) = 2sinleft(π/12tright) - 52 )
C. ( T(t) = -2sinleft(π/6tright) - 52 )
D. ( T(t) = 2cosleft(π/6tright) - 52 )

Answer 1

The formula that models the temperature at the South Pole in March is T(t) = 2 * cos((2π/24) * (t - π/2)) - 52. The temperature at 5 p.m. is approximately -50.63°C.

Step-by-step explanation:

To find the formula of the trigonometric function that models the temperature at the South Pole in March, we need to consider two factors: the highest temperature and the lowest temperature, and their respective times. We know that the highest temperature of about -50°C is reached around 2 p.m. and the lowest temperature of about -54°C is reached half a day apart from the highest temperature, at 2 a.m. To create a cosine function, which has a peak at 2 p.m., we can use the formula T(t) = A * cos(B * (t - C)) + D, where A represents the amplitude, B represents the frequency, C represents the phase shift, and D represents the midline. Since the highest temperature is -50°C, the amplitude A is 2, as the range of temperature change is 4 degrees (from -54°C to -50°C). The frequency B is 2π/24, because it takes 24 hours to complete one full cycle. The phase shift C is π/2, because the highest temperature is reached at 2 p.m., which is halfway through the 24-hour cycle. The midline D is -52, as it is the average of the highest and lowest temperatures (-50°C and -54°C). Therefore, the formula is T(t) = 2 * cos((2π/24) * (t - π/2)) - 52. To find the temperature at 5 p.m., we substitute t = 5 into the formula and calculate the value as follows: T(5) = 2 * cos((2π/24) * (5 - π/2)) - 52. Evaluating this expression, we find that the temperature at 5 p.m. is approximately -50.63°C.