Question

The number of users on a website has grown exponentially since its launch. After 2 months, there were 300 users. After 4 months there were 30000 users. Find the exponential function that models the number of users x months after the website was launched.

Exponential models
ANSWERED
asked 2021-02-21
The number of users on a website has grown exponentially since its launch. After 2 months, there were 300 users. After 4 months there were 30000 users. Find the exponential function that models the number of users x months after the website was launched.

Answers (1)

2021-02-22

Consider the model \(y=a \cdot b^{x}\) which represents the exponential function that models the number of users x months after the website was launched.
Given that after 2 months, there were 300 users and after 4 months there were 30000 users.
This gives the equations
\(\displaystyle{300}={a}\cdot{b}^{{2}}\) (1)
\(\displaystyle{30000}={a}\cdot{b}^{{4}}\) (2)
Divide (2) by (1) and obtain the value of b as follows.
\(\displaystyle{\frac{{{a}\cdot{b}^{{4}}}}{{{a}\cdot{b}^{{2}}}}}={\frac{{{30000}}}{{{300}}}}\)
\(\displaystyle{b}^{{2}}={100}\)
\(\displaystyle{b}={10}\)
Substitute \(b=10\) in (1) and simplify as follows.
\(\displaystyle{300}={a}\cdot{\left({10}\right)}^{{2}}\)
\(\displaystyle{300}={a}\cdot{100}\)
\(\displaystyle{a}={\frac{{{300}}}{{{100}}}}\)
\(\displaystyle{a}={3}\)
Thus, the exponential function that models the number of users x months after the website was launched is \(\displaystyle{y}={3}\cdot{\left({10}\right)}^{{x}}\)

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