Problem 13 - Applied Calculus

Question

Average daily high temperatures in Ottawa, the capital of Canada, range from a low of \(-6^{\circ}\) Celsius on January 1 to a high of \(26^{\circ}\) Celsius on July 1 six months later. See Figure \(1.102 .\) Find a formula for \(H\), the average daily high temperature in Ottawa in, \({ }^{\circ} \mathrm{C}\), as a function of \(t\), the number of months since January \(1 .\)

Step-by-Step Solution

Answer

The formula is \( H(t) = 16 \cos\left(\frac{\pi}{6}t\right) + 10 \).

Step 1: Identify the Given Data

We are given that the average daily high temperature in Ottawa ranges from \(-6^{\circ}\) Celsius in January (\(t = 0\)) to \(26^{\circ}\) Celsius in July (\(t = 6\)). We need to find a periodic function that fits these data points, which suggests using a sinusoidal model.

Step 2: Determine the Amplitude

The amplitude of the sinusoidal function is half of the difference between the maximum and minimum temperatures. Calculate this as follows:\[ \text{Amplitude} = \frac{26 - (-6)}{2} = \frac{32}{2} = 16 \]

Step 3: Determine the Midline

The midline of the sinusoidal function is the average of the maximum and minimum temperatures. Calculate this average:\[ \text{Midline} = \frac{26 + (-6)}{2} = \frac{20}{2} = 10 \]

Step 4: Determine the Period

The period of the function is the number of months from January to the next cycle, which is a full year or 12 months. Thus, the period is 12. The formula for the period of a sinusoid is \( \frac{2\pi}{B} = 12 \), solving for \(B\) gives:\[ B = \frac{2\pi}{12} = \frac{\pi}{6} \]

Step 5: Choose the Appropriate Sine Function

Assume a cosine function which fits the maximum at \(t = 0\). The general form of the sinusoidal function is\[ H(t) = A \cos(Bt + C) + D \]where \(A = 16\), \(B = \frac{\pi}{6}\), \(C = 0\) because the maximum occurs at \(t = 0\), and \(D = 10\).

Step 6: Construct the Final Formula

Plug the values of \(A\), \(B\), and \(D\) into the cosine function:\[ H(t) = 16 \cos\left(\frac{\pi}{6}t\right) + 10 \]This formula models the average daily high temperature in Ottawa as a function of the number of months since January 1.

Key Concepts

Amplitude

To understand how sinusoidal functions model real-world phenomena like temperatures, it's key to grasp the concept of amplitude. In sinusoidal functions, the amplitude is a measure of how much the function value fluctuates above or below its midline. Simply put, it's the height from the midline to the peak.For the temperature data in Ottawa, the amplitude is found by taking half of the range between the highest and lowest temperatures.

• The maximum temperature is 26°C.
• The minimum temperature is -6°C.

Therefore, the amplitude can be calculated as follows:\[ \text{Amplitude} = \frac{26 - (-6)}{2} = \frac{32}{2} = 16 \]This amplitude of 16 tells us that Ottawa's average temperature can deviate up to 16°C above or below the midline, which is the average of the maximum and minimum temperatures.Amplitude is crucial in modeling natural phenomena, as it helps us understand the extent of variation within cycles, such as the swings over the year in the Ottawa temperature model.

Temperature Modeling

Temperature modeling helps us understand how temperatures change over time, making it easier to predict future temperatures based on patterns. We often use sinusoidal functions since they're well-suited to modeling periodic behavior, like seasonal temperature changes.In Ottawa's case, the temperature data from January to July can be effectively captured using a sinusoidal model. This type of function can describe temperatures that belly up to a peak and dip down to a trough in a smooth, predictable pattern - much like the seasons change.When creating a sinusoidal model for temperatures, we use components like:

• Amplitude - To determine how much variation occurs seasonally.
• Midline - This acts like a baseline average temperature.
• Period - Defines how long it takes to complete one cycle (e.g., an entire year).

For Ottawa, using parameters like these helps create a robust and realistic model:\[ H(t) = 16 \cos\left(\frac{\pi}{6}t\right) + 10 \]This model captures how temperatures rise to a peak in summer and fall in winter, aligning with known climatic patterns. By modeling such temperatures, we gain valuable insights into how environmental factors impact our climate from month to month.

Periodic Function

Periodic functions are channels through which we can understand repetitive processes occurring within specific cycles. They return to the same value at regular intervals, making them perfect for representing phenomena that repeat over consistent timeframes like seasons, waves, and even sound patterns.In temperature analysis, a periodic function models how something like the average monthly temperature in Ottawa cycles through weeks and months consistently every year. A sinusoidal function, such as cosine, is an ideal choice because it naturally oscillates with smooth, rhythmic ups and downs, akin to temperature variations throughout the year.Here's what is essential for constructing a periodic function:

• Amplitude - Reflects the magnitude of variation.
• Period - This is the length of one complete cycle, such as 12 months for annual temperature cycles.
• Phase Shift - Determines starting point of the cycle.

For Ottawa's temperature model, the cycle is yearly, returning to the same point every 12 months:\[ B = \frac{2\pi}{12} = \frac{\pi}{6} \]The integration of such periodic functions enriches our understanding of cyclic processes and greatly enhances our predictive capabilities regarding natural and man-made rhythms.