Question

The average number of air conditioners sold in western Canada varies seasonally and depends on the month of the year. The formula \(y=5.9+2.4 \sin \left(\frac{\pi}{6}(t-3)\right)\) gives the expected sales, \(y,\) in thousands, according to the month, \(t,\) where \(t=1\) represents January, \(t=2\) is February, and SO On. a) In what month are sales of 8300 air conditioners expected? b) In what month are sales expected to be least? c) Does this formula seem reasonable? Explain.

Short answer

a) Sales are highest in June. b) Sales are lowest in September. c) The formula is reasonable due to its seasonal variation.

Step-by-step solution

Understand the given formula

The formula provided is \[ y = 5.9 + 2.4 \sin \left(\frac{\pi}{6}(t-3)\right) \] where y represents the sales in thousands, and t represents the month (with t=1 for January, t=2 for February, etc.). Let’s break down each part of the problem.

Solving for part (a)

We need to find the month when y = 8.3 (since 8300 air conditioners equal 8.3 thousands). Plugging y = 8.3 into the formula:\[ 8.3 = 5.9 + 2.4 \sin \left(\frac{\pi}{6}(t-3)\right) \]Subtract 5.9 from both sides:\[ 2.4 = 2.4 \sin \left(\frac{\pi}{6}(t-3)\right) \]Divide both sides by 2.4:\[ 1 = \sin \left(\frac{\pi}{6}(t-3)\right) \]The sine function equals 1 at \(\frac{\pi}{2}\), so solve for t:\[ \frac{\pi}{6}(t-3) = \frac{\pi}{2} \]Multiply both sides by 6/\(\pi\):\[ t-3 = 3 \]Then solve for t:\[ t = 6 \]Therefore, the sales of 8300 air conditioners are expected in month 6, which is June.

Solving for part (b)

The sales expected to be least will be when the value of the sine function is the smallest, which is -1. Plugging this into the formula:\[ y = 5.9 + 2.4 \sin \left(\frac{\pi}{6}(t-3)\right) \]When \sin \left(\frac{\pi}{6}(t-3)\right) = -1, we get:\[ y = 5.9 + 2.4(-1) = 5.9 - 2.4 = 3.5 \]Therefore, we need to find the value of t where the sine is -1. The sine function equals -1 at \(-(\frac{\pi}{2})\), so solve for t:\[ \frac{\pi}{6}(t-3) = -\frac{\pi}{2} \]Multiply both sides by 6/\(\pi\):\[ t-3 = -3 \]Thus,\[ t = 0 \]But since t=0 isn't within our months (1 through 12), we check the next cycle: \[ t+12-3 = 9 \]So the least sales are expected in September, month 9.

Evaluating the Formula

The formula is reasonable if the sales pattern matches typical seasonal air conditioner sales, which are higher in warmer months and lower in colder months. The sine function provides the necessary periodicity to model seasonal variations, confirming the reasonability of the formula.

Key concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The most common trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions are used to model periodic phenomena because they repeat at regular intervals, known as their periods. For example, the sine function has a period of 2π, which means it repeats its values every 2π units. This makes trigonometric functions particularly useful for modeling cyclical events like waves, sound, and yes, seasonal variations on air conditioner sales.

Sine Function

The sine function, denoted as \(\text{sin}(x)\), is one of the basic trigonometric functions. It's defined on the unit circle, where the length of the arc subtended by an angle \(x\) at the center corresponds with the y-coordinate of the point on the circle. For example:

• \(\text{sin}(0) = 0\)
• \(\text{sin}\bigg(\frac{\pi}{2}\bigg) = 1\)
• \(\text{sin}(\pi) = 0\)
• \(\text{sin}\bigg(\frac{3\pi}{2}\bigg) = -1\)

The sine function is periodic with a period of \(2\pi\), meaning it repeats every \({6.28}\) units. Its values oscillate smoothly between \(-1\) and \(1\), making it perfect for representing oscillatory processes like seasons.

Seasonal Variation

Seasonal variations refer to changes or patterns that occur at specific times of the year. These patterns are repetitive every year and are often influenced by the weather, holidays, or other regular events. For instance, the sales of air conditioners typically peak during the summer months and decline during the winter months.
In mathematical modeling, seasonal variations can be captured using trigonometric functions to create periodic models. A common example is the sine function, which can effectively illustrate the ups and downs across seasons. In our exercise, the air conditioner sales peak during a certain month (June) and are at their lowest in another (September), clearly demonstrating seasonal variation.

Mathematical Modeling

Mathematical modeling is the process of representing real-world situations using mathematical equations and concepts. It provides a simplified, abstract representation of complex phenomena and helps us understand, analyze, and make predictions about them.
In the given problem, the equation \(\text{y} = 5.9 + 2.4 \text{sin}\bigg(\frac{\pi}{6} (t - 3)\bigg)\) models the average number of air conditioners sold each month. Breaking it down, we see:

• The constant term \(5.9\) represents the baseline sales.
• The amplitude \(2.4\) of the sine function dictates how much sales can vary from the base.
• The argument of the sine function \(\bigg(\frac{\pi}{6} (t - 3)\bigg)\) determines the timing of the seasonal oscillations.

Using this model, we can accurately predict when sales are highest (June) and lowest (September), reflecting realistic seasonal variation in air conditioner demand.