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Difficulty: MediumLinear and Exponential Growth

A scientist is monitoring the population of two different bacteria cultures, Culture A and Culture B, in a laboratory. The table below shows the cell counts of the two cultures at 11 hour and 33 hours after the start of the experiment (t=0t=0 hours).

Time (tt, hours)Culture A Cell CountCulture B Cell Count
11,2001,200
32,7002,700

It is known that the population of Culture A grows linearly, while the population of Culture B grows exponentially. If A0A_0 is the initial population of Culture A at t=0t=0 hours, and B0B_0 is the initial population of Culture B at t=0t=0 hours, what is the value of B0−A0B_0 - A_0?

  1. A
    150
  2. 350Answer
  3. C
    250
  4. D
    50

Answer

The correct difference between the initial population of Culture B and Culture A is 350.
To find the initial values, we must model both cultures. For the linear growth of Culture A, the population increases by a constant amount each hour. The change over two hours is 2,700−1,200=1,5002,700 - 1,200 = 1,500 cells, which corresponds to a growth rate of 750750 cells per hour. Subtracting this hourly increase from the population at 11 hour (1,2001,200) gives an initial population A0=450A_0 = 450. For the exponential growth of Culture B, the population increases by a constant factor each hour. The ratio of the population at 33 hours to the population at 11 hour is 2,7001,200=2.25\frac{2,700}{1,200} = 2.25, which represents the growth factor squared (b2b^2). Taking the square root gives a growth factor of b=1.5b = 1.5 per hour. Dividing the population at 11 hour (1,2001,200) by this factor gives an initial population B0=800B_0 = 800. The value of B0−A0B_0 - A_0 is 800−450=350800 - 450 = 350.

Step-by-Step Solution

1
Find the constant rate of change and the initial value A0A_0 of the linear model (Culture A).
The hourly rate of change is m=2700−12003−1=750m = \frac{2700 - 1200}{3 - 1} = 750 cells per hour. The initial population is A0=1200−750(1)=450A_0 = 1200 - 750(1) = 450 cells.
Linear growth has a constant absolute increase per unit of time, modeled by A(t)=A0+m⋅tA(t) = A_0 + m \cdot t.
2
Find the constant growth factor and the initial value B0B_0 of the exponential model (Culture B).
The growth factor squared is b2=27001200=2.25  ⟹  b=1.5b^2 = \frac{2700}{1200} = 2.25 \implies b = 1.5. The initial population is B0=12001.51=800B_0 = \frac{1200}{1.5^1} = 800 cells.
Exponential growth has a constant multiplier per unit of time, modeled by B(t)=B0⋅btB(t) = B_0 \cdot b^t.
3
Calculate the difference between the two initial values B0−A0B_0 - A_0.
B0−A0=800−450=350B_0 - A_0 = 800 - 450 = 350.
The question asks for the value of the difference between the initial population of Culture B and the initial population of Culture A.

Key Concept

Linear and Exponential Growth
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Topics

Linear and Exponential Growth

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