Problem 135 - Algebra and Trigonometry Real Mathematics, Real People

Question

The normal daily high temperature \(T\) (in degrees Fahrenheit) in Savannah, Georgia, can be approximated by $$T=76.4+16 \cos \left(\frac{\pi t}{6}-\frac{7 \pi}{6}\right)$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January. Find the normal daily high temperature for each month. (Source: National Climatic Data Center) (a) January (b) July (c) October

Step-by-Step Solution

Answer

The daily high temperatures are calculated as, \n(a) January: \( T \approx 60.4^\circ F\) \n(b) July: \( T \approx 92.4^\circ F\) \n(c) October: \( T \approx 76.4^\circ F\)

Step 1: January Calculation

Substitute the value of \(t=1\) (corresponding to January) into the given formula:\n\n\[T=76.4+16 \cos \left(\frac{\pi (1)}{6}-\frac{7 \pi}{6}\right)\]

Step 2: July Calculation

Substitute the value of \(t=7\) (corresponding to July) into the formula:\n\n\[T=76.4+16 \cos \left(\frac{\pi (7)}{6}-\frac{7 \pi}{6}\right)\]

Step 3: October Calculation

Substitute the value of \(t=10\) (corresponding to October) into the formula:\n\n\[T=76.4+16 \cos \left(\frac{\pi (10)}{6}-\frac{7 \pi}{6}\right)\]

Key Concepts

Trigonometry in Real Life Applications

When we think about the practical applications of trigonometry, most of us relate it to the field of mathematics or engineering. However, trigonometry's reach extends to various real-world scenarios, including meteorology where it is used for weather and climate modeling. The technique can help predict temperatures throughout the year by using trigonometric functions to represent the periodic nature of climate changes.

For example, predictive models for temperature can illustrate the expected highs and lows over the months. By plugging in different values reflecting the time of year, the models provide temperature estimates that help in planning agricultural activities, energy consumption forecasting, and even in the preparation of weather advisories.

Cosine Function

The cosine function is one of the fundamental components of trigonometry. It is used to describe the relationship between the angle of a right triangle and the lengths of its sides. However, outside of geometry, the cosine function serves as a powerful tool to model periodic phenomena such as sound waves, light waves, and yes, even temperature fluctuations.

The cosine function, denoted as 'cos', varies smoothly between -1 and 1, and this characteristic makes it ideal for modeling cyclical events where these maxima and minima play critical roles. It's important to understand its properties and behaviour, especially its period and amplitude, when attempting to create accurate models for real-life applications like temperature predictions.

Temperature Modeling

Temperature modeling involves creating mathematical representations to predict temperature variations over time. These models are crucial for understanding local and global climate patterns. In the textbook exercise, where the temperature in Savannah, Georgia is represented as a function of time with a cosine wave, the importance of defining the period, amplitude, and phase shift of the wave becomes clear. The model showcases how temperatures change cyclically throughout the year, peaking in one period and dipping in another, adjusting these factors will refine the accuracy of the temperature model.

Periodic Functions

Periodic functions are mathematical functions that exhibit a repeating pattern at regular intervals, known as the period. Think of them like the hands of a clock that rotate back to the same position time after time. In trigonometry, the cosine and sine functions are classic examples of periodic functions. They are used to describe oscillations and waves, both in mathematics and in natural phenomena.

The periodic nature of the cosine function, applied in the temperature model for Savannah, mirrors the regular ebb and flow of seasonal temperatures within a year. This repeatable pattern allows for predictions based on time, where each month corresponds to a specific point in the function's cycle. The control over the periodicity is what makes the cosine function so invaluable in creating realistic and dynamic models for various applications.