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Difficulty: MediumLinear and Exponential Growth

A municipal library system models the growth of its e-book collection at two different branches. At the start of the year (t=0t = 0 months), Branch A and Branch B each have 1,0001,000 e-books. The number of e-books at Branch A increases by 150150 e-books each month. The number of e-books at Branch B increases by 20%20\% each month. What is the difference between the number of e-books at Branch B and the number of e-books at Branch A at t=3t = 3 months?

  1. A
    150
  2. 278Answer
  3. C
    378
  4. D
    2,150

Answer

278
The correct answer is obtained by subtracting the number of e-books at Branch A from the number at Branch B at t=3t = 3. Branch A grows linearly: A(3)=1000+150(3)=1450A(3) = 1000 + 150(3) = 1450. Branch B grows exponentially: B(3)=1000(1.2)3=1728B(3) = 1000(1.2)^3 = 1728. The difference is 1728−1450=2781728 - 1450 = 278.

Step-by-Step Solution

1
Define the mathematical functions representing the number of e-books at each branch after tt months.
Branch A (linear growth): A(t)=1000+150tA(t) = 1000 + 150t. Branch B (exponential growth): B(t)=1000(1.2)tB(t) = 1000(1.2)^t.
Establishing the correct equations is necessary to compute the values at a specific time step.
2
Evaluate both functions at t=3t = 3 months.
A(3)=1000+150(3)=1450A(3) = 1000 + 150(3) = 1450. B(3)=1000(1.2)3=1000(1.728)=1728B(3) = 1000(1.2)^3 = 1000(1.728) = 1728.
Calculating the total number of e-books at each branch at the given month allows comparison.
3
Calculate the difference between the number of e-books at Branch B and Branch A.
Difference = 1728−1450=2781728 - 1450 = 278.
Subtracting the linear model output from the exponential model output yields the required difference.

Key Concept

Distinguishing between and evaluating linear growth (constant addition) and exponential growth (constant percentage multiplier).
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Topics

Linear and Exponential Growth

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