Question

In a 366 -day year, the average daily maximum temperature in Vancouver, British Columbia, follows a sinusoidal pattern with the highest value of \(23.6^{\circ} \mathrm{C}\) on day \(208,\) July \(26,\) and the lowest value of \(4.2^{\circ} \mathrm{C}\) on day \(26,\) January 26. a) Use a sine or a cosine function to model the temperatures as a function of time, in days. b) From your model, determine the temperature for day \(147,\) May 26 c) How many days will have an expected maximum temperature of 21.0 ^ \(\mathrm{C}\) or higher?

Short answer

The model function is \( T(t) = 9.7 \cos\left( \frac{2\pi}{366} (t - 208) \right) + 13.9 \). The temperature on day 147 is \( 16.1^{\circ} \mathrm{C} \). There will be approximately 116 days with temperatures 21.0°C or higher.

Step-by-step solution

- Identify the amplitude

The amplitude (A) is half the difference between the maximum and minimum temperatures. So,\[ A = \frac{23.6^{\circ} \mathrm{C} - 4.2^{\circ} \mathrm{C}}{2} = 9.7^{\circ} \mathrm{C} \]

- Determine the vertical shift

The vertical shift (D) is the average of the maximum and minimum temperatures. So,\[ D = \frac{23.6^{\circ} \mathrm{C} + 4.2^{\circ} \mathrm{C}}{2} = 13.9^{\circ} \mathrm{C} \]

- Calculate the period

The period (T) is the number of days in a year, which is 366 days.

- Find the frequency

Frequency (B) is \( \frac{2\pi}{T} \). Hence,\[ B = \frac{2\pi}{366} \]

- Determine the phase shift

For a cosine function, the phase shift (C) corresponds to the day of the maximum temperature, which is 208 days. The phase shift should be negative if the peak is shifted to the right. Thus, \(C = -208\).

- Form the model

With the values calculated, the temperature function can be modeled as follows: \[ T(t) = 9.7 \cos\left( \frac{2\pi}{366} (t - 208) \right) + 13.9 \]

- Find the temperature for day 147

Substitute \( t = 147 \) into the model: \[ T(147) = 9.7 \cos\left( \frac{2\pi}{366} (147 - 208) \right) + 13.9 \] Calculate the cosine value: \[ T(147) = 9.7 \cos\left( \frac{2\pi}{366} (-61) \right) + 13.9 \approx 16.1^{\circ} \mathrm{C} \]

- Calculate days with temperature \( \geq 21.0^{\circ}\mathrm{C}\)

Set \( T(t) \geq 21.0 \) and solve for t: \[ 9.7 \cos\left( \frac{2\pi}{366} (t - 208) \right) + 13.9 \geq 21.0 \] Simplify: \[ \cos\left( \frac{2\pi}{366} (t - 208) \right) \geq \frac{21.0 - 13.9}{9.7} \approx 0.732 \] The cosine function exceeds 0.732 for specific intervals. Without solving here, recognizing it oscillates above this threshold twice per period, each interval of approximately 58 days. Hence, the number of days: \[ 2 \times 58 = 116 \]

Key concepts

Amplitude

The amplitude of a sinusoidal function represents the distance from the middle value to the peak or trough of the wave. For temperature modeling, it tells us how much the temperature fluctuates throughout the year. Specifically, amplitude = half the difference between the highest and lowest temperatures. In this exercise, we calculate it as follows:
\[ A = \frac{23.6^{\circ} \mathrm{C} - 4.2^{\circ} \mathrm{C}}{2} = 9.7^{\circ} \mathrm{C} \]
This 9.7°C indicates how far the temperature goes above and below the average. Understanding the amplitude helps us grasp the extent of seasonal variations.

Vertical Shift

The vertical shift shows how the whole sinusoidal function is moved up or down on the temperature scale. It represents the average temperature over the year. We calculate it by averaging the highest and the lowest temperatures:
\[ D = \frac{23.6^{\circ} \mathrm{C} + 4.2^{\circ} \mathrm{C}}{2} = 13.9^{\circ} \mathrm{C} \]
This value shifts the sinusoidal curve vertically so the average temperature becomes 13.9°C. This is the midpoint between the peak in summer and the trough in winter, making it an essential part of the temperature model.

Frequency

Frequency describes how often the wave completes a cycle in a given time period. For a yearly temperature model, the period is one year (366 days in this case), and the frequency is calculated as:
\[ B = \frac{2\pi}{366} \]
This frequency tells us how quickly the temperature cycle repeats. In simple terms, it determines how stretched or compressed the wave is. In our yearly model, the low frequency corresponds to one complete cycle over 366 days, reflecting the annual repetition of temperature patterns.

Phase Shift

The phase shift indicates how much the wave is shifted horizontally, left or right. It helps align the peak or trough with the given dates. In our task, the highest temperature occurs on day 208, which shifts the cosine function to this day. We calculate the phase shift as:
\[ C = -208 \]
The negative sign shows it is shifted to the right. Incorporating the phase shift into the model ensures that the peak temperature corresponds accurately to day 208, aligning our function with real-world data.

Period

The period is the length of time it takes for the temperature cycle to complete one full cycle. For our temperature model, the period is the number of days in a year. In this exercise, it’s 366 days. This period indicates that temperatures repeat annually, following the same sinusoidal pattern every year. The formula for the function's period is tied to the frequency:
\[ T = 366 \]
Understanding the period helps us know when to expect similar temperature patterns, providing insight into seasonal changes. The period guarantees the model's time-frame matches the actual calendar year.