The economist for a large sporting-goods manufacturer developed the following function to model the company's sales, where S is sales in millions of dollars and t is the week of the year, beginning January 1 of each year:
S = 8 + t/52 – 6cos(πt/26)
Which of the following accurately describes the sales cycle over several years?
A.)Short-term cycles with short-term decline
B.)Long-term cycles with long-term decline
C.)Long-term cycles with short-term growth
D.)Short-term cycles with long-term growth
If we look at the cosine function which can be written as
cos((t/52)2π), we conclude the cycles are over periods of 52 weeks, meaning yearly cycles.
The first two terms represent a straight line with a y-intercept of 8, and constantly increasing at the rate of (1/52).
Can you deduce the answer from the above information?
I'm thinking long term cycles with short term growth?
To analyze the sales cycle over several years, we can observe the behavior of the function over a longer period of time.
Looking at the function S = 8 + t/52 - 6cos(πt/26), we can break it down into its components:
- The term "8" represents a constant base level of sales. This implies that there is a minimum level of sales regardless of the week.
- The term "t/52" increases sales linearly with time, with 52 representing the number of weeks in a year. This indicates a gradual and consistent increase in sales over time.
- The term "-6cos(πt/26)" represents a cyclical pattern in sales. The cosine function produces a periodic wave, with the amplitude of the wave being 6. The period of the wave is given by πt/26, which means it completes one full cycle every 26 weeks.
Based on this analysis, we can conclude that the sales cycle described by the function shows both short-term cycles (as indicated by the cosine wave) and long-term growth (as indicated by the linear term t/52). Therefore, the correct answer is D.) Short-term cycles with long-term growth.
To understand the sales cycle described by the function, we need to analyze its components. Let's break down the function:
S = 8 + t/52 – 6cos(πt/26)
The function consists of three parts:
1. The constant term, 8, represents the base sales value.
2. The term t/52 represents the increase in sales linearly over time. Since there are 52 weeks in a year, this term contributes to the long-term growth of sales.
3. The term – 6cos(πt/26) represents the cyclical nature of sales and is affected by the cosine function. The cosine function oscillates between -1 and 1 as t changes. The period of the cosine function is determined by the coefficient of t, which in this case is π/26. The coefficient π/26 indicates that the sales cycle repeats approximately every 26 weeks or half a year.
Now, let's analyze the options:
A.) Short-term cycles with short-term decline: This option implies that the sales cycles are short, and there is a decrease in sales in the short term. However, the given function does not show any decline throughout the expression.
B.) Long-term cycles with long-term decline: This option implies that the sales cycles are long, and there is a long-term decline in sales. No evidence suggests a decline in the given function.
C.) Long-term cycles with short-term growth: This option suggests that the sales cycles are long, and there is short-term growth. The given function does show long-term growth (t/52), but it does not explicitly show short-term growth. Therefore, this option is not accurately described.
D.) Short-term cycles with long-term growth: This option suggests that the sales cycles are short, and there is long-term growth. The given function does show long-term growth (t/52), but it does not explicitly exhibit short-term cycles. Hence, this option is not accurately described either.
Based on the analysis above, none of the options accurately describes the sales cycle over several years according to the given function.