The number of subscribers, [tex]$f(t)$[/tex], to a magazine after [tex]$t$[/tex] years is shown by the equation below:
[tex]$f(t)=95(0.75)^t$[/tex]
Which conclusion is correct about the number of subscribers to the magazine?
A. It increased by 25% every year.
B. It decreased by 25% every year.
C. It increased by 75% every year.
D. It decreased by 75% every year.
Answer (2)
Calculate the ratio of subscribers in year t + 1 to year t : f ( t ) f ( t + 1 ) = 95 ( 0.75 ) t 95 ( 0.75 ) t + 1 .
Simplify the ratio: f ( t ) f ( t + 1 ) = 0.75 .
Calculate the percentage change: ( 0.75 − 1 ) × 100 = − 25% .
Conclude that the number of subscribers decreases by 25% every year: It decreased by 25% every year.
Explanation
Understanding the Problem The problem provides the function f ( t ) = 95 ( 0.75 ) t which models the number of subscribers to a magazine after t years. We need to determine whether the number of subscribers increases or decreases each year and by what percentage.
Finding the Ratio To determine the change in the number of subscribers each year, we can compare the number of subscribers in year t + 1 to the number of subscribers in year t . This can be done by finding the ratio f ( t ) f ( t + 1 ) .
Calculating the Ratio We have f ( t ) = 95 ( 0.75 ) t and f ( t + 1 ) = 95 ( 0.75 ) t + 1 . Therefore, the ratio is: f ( t ) f ( t + 1 ) = 95 ( 0.75 ) t 95 ( 0.75 ) t + 1 .
Simplifying the Ratio Simplifying the ratio, we get: f ( t ) f ( t + 1 ) = 95 ( 0.75 ) t 95 ( 0.75 ) t + 1 = ( 0.75 ) t + 1 − t = 0.75
Calculating the Percentage Change Since the ratio is 0.75, this means that the number of subscribers each year is 0.75 times the number of subscribers the previous year. To find the percentage change, we calculate ( 0.75 − 1 ) × 100 = − 0.25 × 100 = − 25% .
Conclusion A percentage change of -25% indicates that the number of subscribers decreases by 25% each year.
Final Answer Therefore, the correct conclusion is that the number of subscribers decreased by 25% every year.
Examples
Consider a scenario where a local newspaper's subscription rate is declining. The model f ( t ) = 95 ( 0.75 ) t can help predict how many subscribers the newspaper will have in the future if the decline continues at the same rate. Understanding this decline can help the newspaper make strategic decisions to retain subscribers or adjust its business model. For example, after 1 year, the number of subscribers would be 95 ( 0.75 ) 1 = 71.25 , and after 2 years, it would be 95 ( 0.75 ) 2 = 53.4375 . This model helps in forecasting and planning.
The number of subscribers to the magazine decreases by 25% every year. This is determined by the function f ( t ) = 95 ( 0.75 ) t , where the ratio of subscribers year over year shows a decline. The correct choice is B: It decreased by 25% every year.
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