Problem 59 - Contemporary Precalculus

Question

The average monthly temperature in Cleveland, Ohio is approximated by $$f(t)=22.7 \sin (.52 x-2.18)+49.6$$ where \(t=1\) corresponds to January, \(t=2\) to February, and so on. (a) Construct a table of values \((t=1,2, \ldots, 12)\) for the function \(f(t)\) and another table for \(f(t+12.083)\) (b) Based on these tables would you say that the function \(f\) is (approximately) periodic? If so, what is the period? Is this reasonable?

Step-by-Step Solution

Answer

Answer: Based on the differences of values, the function \(f\) is approximately periodic. The period of the function is approximately \(12.083\) months. This result is reasonable since it aligns with the annual temperature cycle.

Step 1: Compute the function values for each month

We have to compute \(f(t)\) for \(t = 1, 2, \ldots, 12\) and \(f(t+12.083)\) as well. Let's start by computing the values of \(f(t)\) and \(f(t+12.083)\) for each month: Table for \(f(t)\): | Month (t) | \(f(t)\) | |-----------|-------------| | 1 | \(f(1)\) | | 2 | \(f(2)\) | | 3 | \(f(3)\) | | 4 | \(f(4)\) | | 5 | \(f(5)\) | | 6 | \(f(6)\) | | 7 | \(f(7)\) | | 8 | \(f(8)\) | | 9 | \(f(9)\) | | 10 | \(f(10)\) | | 11 | \(f(11)\) | | 12 | \(f(12)\) | Table for \(f(t+12.083)\): | Month (t) | \(f(t+12.083)\) | |-----------|---------------| | 1 | \(f(13.083)\) | | 2 | \(f(14.083)\) | | 3 | \(f(15.083)\) | | 4 | \(f(16.083)\) | | 5 | \(f(17.083)\) | | 6 | \(f(18.083)\) | | 7 | \(f(19.083)\) | | 8 | \(f(20.083)\) | | 9 | \(f(21.083)\) | | 10 | \(f(22.083)\) | | 11 | \(f(23.083)\) | | 12 | \(f(24.083)\) | The student needs to compute the function values for each month and fill in the tables above.

Step 2: Analyze the tables for periodicity

First, the student needs to examine the differences between the values of \(f(t)\) and \(f(t+12.083)\) for each month. If these differences are relatively small (close to zero), we can consider that the function is approximately periodic. If the function is indeed approximately periodic, we need to find the period of the function. Since we have used \(t+12.083\) in the second table, we can approximate the period as \(12.083\) months.

Step 3: Determine if the period is reasonable

Now, the student should think about the physical meaning behind the period of the function. In this context, the period represents the number of months required for the temperature cycle to repeat itself – that is, the annual cycle. Since our approximated period is very close to 12 months, this seems to be a reasonable result, as it aligns with the expected annual cycle of temperature due to the Earth's revolution around the Sun. So, the student concludes: 1. Based on the differences of values, the function \(f\) is approximately periodic. 2. The period of the function is approximately \(12.083\) months. 3. This result is reasonable since it aligns with the annual temperature cycle.

Key Concepts

Periodicity of Trigonometric Functions

Periodicity is a fundamental concept in trigonometry, which is often visualized as the regular repeating of values in a function over a certain interval known as the period. Specifically, trigonometric functions like the sine and cosine have a periodic nature, meaning they repeat their values at regular intervals. A function, f, is said to be periodic if there exists a positive number, P, such that for all values of the variable, x, in the domain of f, the equation f(x) = f(x + P) holds true. The smallest positive number, P, for which this is true is the period of the function.

For example, the sine function, sin(x), has a period of 2π radians or 360 degrees. This implies that the sine function will repeat its pattern every 2π radians. This periodicity is inherently related to the circular nature of trigonometric functions, as they are defined around the unit circle. When analyzing a trigonometric function for periodicity, evaluating at intervals equal to the suspected period can show whether the function values repeat.

Understanding the periodicity of these functions is crucial for many applications, including signal processing, sound waves, and the natural cycles observed in real-life scenarios, such as the problem of modeling temperature throughout the year in a specific location.

Sine Function Applications

The sine function, with its perfectly cyclical nature, finds application in a multitude of fields. Its pattern of repeating values at regular intervals makes it an excellent choice for modeling phenomena that exhibit periodic behavior. One of the sine function's most prevalent applications is in the field of physics, particularly in wave mechanics, where it is used to describe oscillatory motion such as sound waves, light waves, and alternating electric currents.

Moreover, because of its periodic nature, the sine function is also invaluable in electrical engineering for analyzing alternating current (AC) circuits. In geography and climatology, the sine function helps in modeling seasonal temperature variations, daylight hours, and tides, which are all periodic in nature. Even in economics, the sine function can be employed to model repeated trends or cycles over time.

In essence, the sine function's relevance spans from the most basic physics classes to advanced research in various scientific fields, illustrating the wide range of real-life applications hinging on this fundamental trigonometric function.

Modeling Temperature Cycles

Modeling real-world temperature cycles is a central application of the sine function in the realm of mathematics. Due to the Earth's axial tilt and its orbit around the Sun, the temperature in any given location shows a periodic cycle, peaking at certain times of the year and reaching a minimum at others. The sine function, with its periodic properties, can elegantly represent these seasonal temperature changes.

For instance, a basic model for average monthly temperature, like the one in our exercise, could take the form f(t) = A sin(Bt - C) + D, where A represents the amplitude (the height of the wave from its average value), B relates to the period of the cycle, C is the phase shift (how much the cycle shifts horizontally), and D indicates the baseline or average temperature around which the values oscillate.

The exercise requires analyzing the periodicity by constructing a table for one yearly cycle and then for one slightly longer than a year (t+12.083). By comparing these tables, students can determine that the temperature cycle is indeed periodic, with a period closely approximating a year, affirming the predictability of seasonal temperature variations. This approach not only demonstrates the practical use of trigonometry but also how mathematics can help us understand and predict natural phenomena.