Success in business depends on many factors. Buying habits of consumers, trends in fashion and taste, seasonal trends, the strength of the competition, and the condition of the national and international economies all affect the fortunes of a business. The result of these factors is that businesses often show a broad overall pattern of growth or decline superimposed on a continuing series of small ups and downs. Such business cycles can often be modeled by using trigonometric functions. Recall that sinusoidal trigonometric functions are periodic.

Question

Success in business depends on many factors. Buying habits of consumers, trends in fashion and taste, seasonal trends, the strength of the competition, and the condition of the national and international economies all affect the fortunes of a business. The result of these factors is that businesses often show a broad overall pattern of growth or decline superimposed on a continuing series of small ups and downs. Such business cycles can often be modeled by using trigonometric functions. Recall that sinusoidal trigonometric functions are periodic.

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 terms in the equation increases as t increases?

cos(πt/26)
-6cos(πt/26)
t/52
8

Answer 1

you know that cos(u) might either decrease or increase, depending on u.

surely t/52 increases when t does.

!!

Answer 2

To determine which term in the equation increases as t increases, we need to understand the effect of each term on the equation.

1. cos(πt/26): This term represents a cosine function of πt/26, where t is the week of the year. As t increases, the cosine function oscillates between -1 and 1. When t increases, the value of cos(πt/26) may increase or decrease depending on the specific value of t, but it does not exhibit a consistent increasing or decreasing pattern.

2. -6cos(πt/26): This term is derived from cos(πt/26) by multiplying it by -6. Multiplying by -6 reflects the cosine values such that the amplitude is multiplied by -6. Therefore, as t increases, the value of this term decreases.

3. t/52: This term represents the linear increase in sales over time. Since t represents the week of the year, as t increases, the numerator of t/52 increases, resulting in a larger value for the entire fraction. Therefore, as t increases, the value of this term increases.

4. 8: This term is a constant and does not depend on the value of t. It remains constant at 8 regardless of the value of t.

From the analysis above, we can conclude that the term t/52 increases as t increases.