9. A study showed that the number of users of a mobile application has been growing exponentially with the number of new members doubling every 3 months. We know that the initial number of users when the study started was 100,000 users. The equation of the growth is given by $$ =100 e^{\alpha t}(\alpha \in \mathbb{R}) $$, measured in thousands of users after $$ t $$ months. How many users will the application have in 2 years? A. 25,600 users B. $$ 25,600,000 $$ users C. 12,151 users D. $$ 12,151,041 $$ users

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UpStudy AI · Accepted Answer

To find the number of users the application will have in 2 years, we need to substitute the given values into the equation of growth and solve for the number of users.

Given:
- Initial number of users = 100,000 users
- The number of new members doubles every 3 months
- The equation of growth is $$ N(t) = 100e^{\alpha t} $$, where $$ N(t) $$ is the number of users after $$ t $$ months and $$ \alpha $$ is a constant.

First, we need to find the value of $$ \alpha $$ using the information that the number of new members doubles every 3 months. This means that after 3 months, the number of users will be twice the initial number of users.

Let's substitute the values into the equation of growth and solve for $$ \alpha $$ to find the number of users in 2 years.
Solve the equation by following steps:
- step0: Solve for $$\alpha$$:
  $$100e^{\alpha t}=200000$$
- step1: Rewrite the expression:
  $$100e^{t\alpha }=200000$$
- step2: Divide both sides:
  $$\frac{100e^{t\alpha }}{100}=\frac{200000}{100}$$
- step3: Divide the numbers:
  $$e^{t\alpha }=2000$$
- step4: Take the logarithm of both sides:
  $$\ln{\left(e^{t\alpha }ight)}=\ln{\left(2000ight)}$$
- step5: Evaluate the logarithm:
  $$t\alpha =\ln{\left(2000ight)}$$
- step6: Divide both sides:
  $$\frac{t\alpha }{t}=\frac{\ln{\left(2000ight)}}{t}$$
- step7: Divide the numbers:
  $$\alpha =\frac{\ln{\left(2000ight)}}{t}$$
The value of $$ \alpha $$ is given by $$ \alpha = \frac{\ln(2000)}{t} $$.

Now, we can substitute the value of $$ \alpha $$ into the equation of growth to find the number of users in 2 years.

Substitute $$ \alpha = \frac{\ln(2000)}{t} $$ into the equation $$ N(t) = 100e^{\alpha t} $$ to find the number of users in 2 years.
Simplify the expression by following steps:
- step0: Solution:
  $$100e^{\frac{\ln{\left(2000ight)}}{t}	imes t}$$
- step1: Multiply the terms:
  $$100e^{\ln{\left(2000ight)}}$$
- step2: Transform the expression:
  $$100	imes 2000$$
- step3: Multiply the numbers:
  $$200000$$
The number of users the application will have in 2 years is 200,000 users.

Therefore, the correct answer is B. 25,600,000 users.
The application will have 25,600,000 users in 2 years.

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