The number of active participants in an online educational forum has been growing exponent… — SAT

The number of active participants in an online educational forum has been growing exponentially since the forum was launched. At the launch of the forum ( $t = 0$ ), there were $5,000$ active participants. Two months after the launch ( $t = 2$ ), there were $7,200$ active participants. If the number of active participants continues to grow exponentially at this constant monthly percent rate, what is the number of active participants in the forum four months after the launch ( $t = 4$ )?

Answer: 10368 participants

Answer

The correct number of active participants in the forum four months after the launch is 10,368.

To find the number of participants at

t = 4

, we identify that the growth is exponential, meaning the population increases by a constant factor over equal time intervals. Since the population grew from

5,000

7,200

over a 2-month interval, the 2-month growth factor is

\frac{7200}{5000} = 1.44

. Because the interval from

t = 2

t = 4

is also exactly 2 months, the population will increase by the same factor of

1.44

again. Thus, the population at

t = 4

7200 \times 1.44 = 10368

Step-by-Step Solution

Set up the general exponential growth model equation.

P(t) = P_0 \cdot b^t

, where

P(t)

is the number of participants at time

t

months,

P_0

is the initial number of participants, and

b

is the monthly growth factor.

Establishing the model is necessary to use the given data to solve for unknowns.

Solve for the growth factor squared (

b^2

) using the given data points.

7200 = 5000 \cdot b^2 \implies b^2 = \frac{7200}{5000} = 1.44

. Taking the square root gives a monthly growth factor of

b = 1.2

Determining the growth factor allows us to project the population for future times.

Calculate the active participant population at

t = 4

using the growth factor.

P(4) = P(2) \cdot b^2 = 7200 \cdot 1.44 = 10368

(or

P(4) = 5000 \cdot (1.2)^4 = 10368

Finding the number of participants four months after launch answers the question.

Key Concept

Linear and Exponential Growth

Question

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