Problem 136
Question
A company that produces wakeboards forecasts monthly sales \(S\) during a two- year period to be $$S=2.7+0.142 t+2.2 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right)$$ where \(S\) is measured in hundreds of units and \(t\) is the time (in months), with \(t=1\) corresponding to January \(2014 .\) Estimate sales for each month. (a) January 2014 (b) February 2015 (c) May 2014 (d) June 2015
Step-by-Step Solution
Verified Answer
The estimated sales for each given month are as follows:(a) January 2014: The sales S would be calculated after substituting t=1 in the given equation.(b) February 2015: The sales S would be calculated after substituting t=13 in the given equation.(c) May 2014: The sales S would be calculated after substituting t=5 in the given equation.(d) June 2015: The sales S would be calculated after substituting t=18 in the given equation. These estimated sales need to be simplified to get the final result.
1Step 1: Understanding the Sales Function
Firstly, on understanding the function S=2.7+0.142t+2.2sin((πt/6)-(π/2)), it can be seen that 't' represents time in months and 'S' represents sales in hundreds of units. This equation will be used to calculate the sales for the specific months asked in the problem.
2Step 2: Calculation of sales for January 2014
Substitute t=1 (January 2014) into the sales function. S=2.7 + 0.142(1) + 2.2sin((π*1/6)-(π/2)). Simplify to get the estimated sales.
3Step 3: Calculation of sales for February 2015
Substitute t=13 (February 2015) into the sales function. S=2.7 + 0.142(13) + 2.2sin((π*13/6)-(π/2)). Simplify to get the estimated sales.
4Step 4: Calculation of sales for May 2014
Substitute t=5 (May 2014) into the sales function. S=2.7 + 0.142(5) + 2.2sin((π*5/6)-(π/2)). Simplify to get the estimated sales.
5Step 5: Calculation of sales for June 2015
Substitute t=18 (June 2015) into the sales function. S=2.7 + 0.142(18) + 2.2sin((π*18/6)-(π/2)). Simplify to get the estimated sales.
Key Concepts
Sinusoidal Function ApplicationTrigonometric Functions in Real LifePeriodic Functions in EconomicsAlgebraic Problem-Solving
Sinusoidal Function Application
When dealing with real-world problems in various fields such as physics, engineering, and economics, we frequently encounter patterns that repeat themselves over time. In mathematics, these repeating patterns are best described using sinusoidal functions, which are a subset of trigonometric functions.
Sinusoidal functions, like the one in our exercise \(S=2.7+0.142t+2.2 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right)\), are particularly useful in modeling periodic phenomena. In our case, the company's monthly sales, denoted by \(S\), are predicted using a function that fluctuates sinusoidally over time represented by \(t\). This mirrors how sales might increase to a peak and then decrease cyclically with seasons or consumer trends, which commonly happens with products like wakeboards that may be more popular in certain periods of the year.
Sinusoidal functions, like the one in our exercise \(S=2.7+0.142t+2.2 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right)\), are particularly useful in modeling periodic phenomena. In our case, the company's monthly sales, denoted by \(S\), are predicted using a function that fluctuates sinusoidally over time represented by \(t\). This mirrors how sales might increase to a peak and then decrease cyclically with seasons or consumer trends, which commonly happens with products like wakeboards that may be more popular in certain periods of the year.
Trigonometric Functions in Real Life
Trigonometric functions are not just tools for solving triangles in geometry class; they hold significant significance in real-life situations. For instance, the sine wave—a fundamental trigonometric function akin to the one used in our sales forecasting model—is essential in understanding and modeling phenomena such as sound waves, light waves, and even stock market analysis.
In dealing with business sales data, trigonometric functions model cyclical behaviors, where the amplitude signifies the strength of the seasonality, the period reveals how long a full sales cycle lasts, and the phase shift indicates at what point in time a sales cycle commences.
In dealing with business sales data, trigonometric functions model cyclical behaviors, where the amplitude signifies the strength of the seasonality, the period reveals how long a full sales cycle lasts, and the phase shift indicates at what point in time a sales cycle commences.
Periodic Functions in Economics
The use of periodic functions in economics allows analysts to predict behaviors and trends that recur over set intervals. A sine function, as applied in our exercise for sales forecasting, is an excellent tool for representing phenomena with regularity, such as seasonal sales patterns, economic cycles, or market fluctuations.
Economic variables, such as the demand for a product, often exhibit seasonality which can be captured effectively by the regular peaks and troughs of a sinusoidal model. The coefficients and parameters within such a model can be intricately adjusted to reflect changes specific to the economic environment or consumer behaviors.
Economic variables, such as the demand for a product, often exhibit seasonality which can be captured effectively by the regular peaks and troughs of a sinusoidal model. The coefficients and parameters within such a model can be intricately adjusted to reflect changes specific to the economic environment or consumer behaviors.
Algebraic Problem-Solving
Algebra is a field of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. The essence of algebraic problem-solving lies in the ability to manipulate these symbols to solve for unknown variables.
In the exercise provided, problem-solving involves identifying the correct time period (\(t\)) to be substituted into the sales forecasting equation and then solving that equation to find the projected sales (\(S\)). Whether it's for January 2014 or June 2015, the process of substituting the appropriate month as 't' and following algebraic operations to reduce the complex equation down to a single sales figure showcases practical algebra at work. Problem solvers must pay careful attention to each part of the equation, as each parameter—the linear trend, the amplitude, the period, and the phase shift—serves a purpose in projecting the sales figure accurately.
In the exercise provided, problem-solving involves identifying the correct time period (\(t\)) to be substituted into the sales forecasting equation and then solving that equation to find the projected sales (\(S\)). Whether it's for January 2014 or June 2015, the process of substituting the appropriate month as 't' and following algebraic operations to reduce the complex equation down to a single sales figure showcases practical algebra at work. Problem solvers must pay careful attention to each part of the equation, as each parameter—the linear trend, the amplitude, the period, and the phase shift—serves a purpose in projecting the sales figure accurately.
Other exercises in this chapter
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