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Difficulty: MediumLinear and Exponential Growth

An online streaming service tracks the number of subscribers in two regions, Region X and Region Y. At the beginning of 2020 (t=0t = 0), Region X had 12,00012,000 subscribers, and the number of subscribers in Region X increased by a constant amount of 2,5002,500 each year. Region Y had 8,0008,000 subscribers at the beginning of 2020, and the number of subscribers in Region Y increased by a constant 15%15\% each year. What is the positive difference between the number of subscribers in Region X and the number of subscribers in Region Y at the beginning of 2022 (t=2t = 2)?

Answer: 6420

Answer

The positive difference between the subscriber counts in Region X and Region Y at the beginning of 2022 is 6,4206,420.
To find the positive difference, we first calculate the subscriber counts for both regions at t=2t = 2. Region X grows linearly at a constant rate of 2,5002,500 subscribers per year, so its population at t=2t = 2 is 12,000+2,500(2)=17,00012,000 + 2,500(2) = 17,000. Region Y grows exponentially at a constant rate of 15%15\% per year, so its population at t=2t = 2 is 8,000×(1.15)2=10,5808,000 \times (1.15)^2 = 10,580. The positive difference between the subscriber counts is 17,000−10,580=6,42017,000 - 10,580 = 6,420.

Step-by-Step Solution

1
Calculate the subscriber count for Region X at t=2t = 2.
17,00017,000
Since Region X grows linearly by a constant amount of 2,5002,500 subscribers per year, the number of subscribers at t=2t = 2 is given by 12,000+2,500(2)=17,00012,000 + 2,500(2) = 17,000.
2
Calculate the subscriber count for Region Y at t=2t = 2.
10,58010,580
Since Region Y grows exponentially by a constant rate of 15%15\% per year, the number of subscribers at t=2t = 2 is given by 8,000×(1.15)2=10,5808,000 \times (1.15)^2 = 10,580.
3
Calculate the positive difference between the two subscriber counts.
6,4206,420
Subtract the smaller subscriber count from the larger subscriber count: 17,000−10,580=6,42017,000 - 10,580 = 6,420.

Key Concept

Distinguishing between linear growth (constant rate of change per unit of time) and exponential growth (constant percent rate of change per unit of time) to solve real-world problems.
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Topics

Linear and Exponential Growth

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