Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

The monthly unit sales \(U\) (in hundreds) of skis for a chain of sports stores are approximated by \(U=58.3+32.5 \cos \frac{\pi t}{6}\) where \(t\) is the time (in months), with \(t=1\) corresponding to January. Determine the months in which unit sales exceed 7500

Step-by-Step Solution

Answer

After rounding the solutions to the nearest whole number and aligning with the calendar months, it will be found that the unit sales exceed 7500 in the specific months.

Step 1: Analyze and Set Up the Inequality

Since 'U' is given in hundreds, then sales exceed 7500 when 'U' exceeds 75. Thus we are looking to solve the following inequality: \( U = 58.3 + 32.5 \cos \frac{\pi t}{6} > 75 \). Now our goal is to isolate the cosine term and create an equation.

Step 2: Isolate the Cosine Term and Create an Equation

First, isolate the cosine term by subtracting 58.3 from both sides of the inequality: \( 32.5 \cos \frac{\pi t}{6} > 75 - 58.3 \). Now divide both sides by 32.5: \( \cos \frac{\pi t}{6} > \frac{75 - 58.3}{32.5} \). Transform this inequality into an equation that we can solve for 't': \( \cos \frac{\pi t}{6} = \frac{75 - 58.3}{32.5} \)

Step 3: Solve the Equation for 't'

Using the cosine inverse function to solve for 't', we get: \( t = 6 \times \frac{\cos^{-1} \left( \frac{75 - 58.3}{32.5} \right)}{\pi} \). Remember cosine function is positive in both the first and fourth quadrant, therefore add the period to find another solution.

Step 4: Obtain Month Values

We use the periodicity of the cosine function and add the period 12 months until we exceed 12. We round our answers to whole numbers because we are dealing with months. We remember that 't=1' corresponds to January, so we align our answers to calendar months.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry. It is used to describe oscillating phenomena like waves. In the equation given, the cosine function helps model the fluctuations in ski sales throughout the year.

The cosine function's range is from -1 to 1, creating a wave-like pattern. When combined with other numbers, like in this problem, it scales and shifts the wave. The equation is part of a cosine curve that represents how sales change monthly.

Understanding the cosine function helps us predict periods of popularity, such as in colder months when people buy more skis.

Inequality Solving

Inequalities are mathematical expressions showing that one quantity is larger or smaller than another. In this exercise, we're tasked with finding when ski sales exceed 7500 units, which means solving an inequality.

The original inequality is manipulated to isolate the cosine term. By performing calculations like subtracting and dividing, we turn the inequality into a more manageable form. This lets us determine when the cosine part reaches a specific value that leads to exceeding the target sales.

Solving inequalities is key not just in math but also in real-world situations like budgeting and planning.

Periodic Functions

Periodic functions repeat their values in regular intervals. The cosine function is periodic with a well-known interval or period, making it ideal for modeling repeating patterns like seasonal sales.

In this problem, since the cosine function has a natural period of 2π, we apply this knowledge to determine when sales thresholds are crossed multiple times within its cycle.

This periodic nature allows us to calculate sales fluctuations accurately across the months of the year, highlighting peaks and troughs as seasons change.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the ratio of sides in a right triangle is known. Here, we use the inverse cosine function to solve for the variable 't', representing months.

By isolating the cosine term and then applying the inverse cosine, we can find specific points where the cosine function meets certain conditions, like exceeding a sales figure.

This approach helps interpret data in real terms, such as translating mathematical solutions into months of the year, providing a tangible understanding of when sales goals are reached.