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Difficulty: MediumLinear and Exponential Growth

At the start of 20182018, City A and City B each recycled 12,00012,000 tons of plastic. For the next several years, the annual amount of plastic recycled in City A increased by 800800 tons each year. During the same period, the annual amount of plastic recycled in City B increased by 5%5\% each year. Which of the following is closest to the difference, in tons, between the annual amount of plastic recycled in City B and the annual amount of plastic recycled in City A at the start of 20232023?

  1. 685Answer
  2. B
    719
  3. C
    1,000
  4. D
    1,485

Answer

685 tons
The correct answer is 685. At the start of 2023, 5 years have elapsed since the start of 2018. City A's plastic recycling volume is modeled by a linear equation: 12,000+800(5)=16,00012,000 + 800(5) = 16,000 tons. City B's plastic recycling volume is modeled by an exponential equation: 12,000(1.05)5≈15,31512,000(1.05)^5 \approx 15,315 tons. The difference between these two volumes is 16,000−15,315=68516,000 - 15,315 = 685 tons.

Step-by-Step Solution

1
Determine the time interval from the start of 2018 to the start of 2023.
t=5t = 5 years
Calculating the difference in years between 2023 and 2018 gives the number of years of growth elapsed.
2
Model and calculate the amount of recycled plastic for City A at t=5t = 5.
A(5)=12,000+800(5)=16,000A(5) = 12,000 + 800(5) = 16,000 tons
City A experiences linear growth, increasing by a constant amount of 800 tons each year.
3
Model and calculate the amount of recycled plastic for City B at t=5t = 5.
B(5)=12,000(1+0.05)5=12,000(1.05)5≈15,315.38B(5) = 12,000(1 + 0.05)^5 = 12,000(1.05)^5 \approx 15,315.38 tons
City B experiences exponential growth, increasing by a constant percentage of 5% each year.
4
Calculate the difference between the recycling amounts of the two cities.
∣16,000−15,315.38∣≈684.62|16,000 - 15,315.38| \approx 684.62 tons
Subtracting the exponential growth value from the linear growth value yields the difference, which rounds to 685 tons.

Key Concept

Comparing linear and exponential growth models over a specified time interval.
Estimated Time:1m 30s
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Topics

Linear and Exponential Growth

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