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Question

Difficulty: MediumLinear and Exponential Growth

The table below shows the insect populations in two different greenhouses, Greenhouse P and Greenhouse Q, monitored over a period of several weeks. The population in Greenhouse P grows linearly, while the population in Greenhouse Q grows exponentially.

Time (weeks)Greenhouse P populationGreenhouse Q population
0500500
1600600
2700720

Based on the table, what is the positive difference between the population in Greenhouse Q and the population in Greenhouse P at week 3?

  1. A
    20
  2. B
    40
  3. 64Answer
  4. D
    164

Answer

64
At week 3, the population of Greenhouse P (which grows by a constant 100 insects each week) is 500+100(3)=800500 + 100(3) = 800. The population of Greenhouse Q (which grows by a constant factor of 1.2 each week) is 500(1.2)3=864500(1.2)^3 = 864. The positive difference between the two populations is 864−800=64864 - 800 = 64.

Step-by-Step Solution

1
Determine the linear growth model for the population in Greenhouse P.
The population in Greenhouse P starts at 500 at week 0 and increases to 600 at week 1, and 700 at week 2. This represents a constant increase of 100 insects per week. Therefore, the population of Greenhouse P at week 3 is 700+100=800700 + 100 = 800 (or 500+100(3)=800500 + 100(3) = 800).
To find the population of Greenhouse P at week 3, we must determine and apply its constant rate of change.
2
Determine the exponential growth model for the population in Greenhouse Q.
The population in Greenhouse Q starts at 500 at week 0 and increases to 600 at week 1, and 720 at week 2. The ratio between consecutive weeks is 600500=1.2\frac{600}{500} = 1.2 and 720600=1.2\frac{720}{600} = 1.2. This represents a constant growth factor of 1.2 per week. Therefore, the population of Greenhouse Q at week 3 is 720×1.2=864720 \times 1.2 = 864 (or 500×(1.2)3=864500 \times (1.2)^3 = 864).
To find the population of Greenhouse Q at week 3, we must determine and apply its constant growth factor.
3
Calculate the positive difference between the two populations at week 3.
The positive difference is 864−800=64864 - 800 = 64.
The question asks for the positive difference between the population of Greenhouse Q and the population of Greenhouse P at week 3.

Key Concept

Linear and Exponential Growth
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Topics

Linear and Exponential Growth

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