Answer :
Sure, let's tackle this problem step-by-step.
### Step 1: Understanding the Problem
We are given:
- The initial number of infected people, [tex]\( N_0 = 400 \)[/tex].
- The growth rate of the infection per day, [tex]\( r = 25\% = 0.25 \)[/tex].
- The duration to predict the number of infections, [tex]\( t = 10 \)[/tex] days.
We need to find how many people will be infected after 10 days.
### Step 2: Writing the Exponential Growth Function
The situation describes exponential growth, where the number of infected people grows by a fixed percentage each day. The general form of an exponential growth function is:
[tex]\[ N(t) = N_0 \cdot (1 + r)^t \][/tex]
Where:
- [tex]\( N(t) \)[/tex] is the number of infected people at time [tex]\( t \)[/tex].
- [tex]\( N_0 \)[/tex] is the initial number of infected people.
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time period (in days in this case).
### Step 3: Applying the Values to the Function
Let's apply the given values to the exponential growth function:
[tex]\[ N(10) = 400 \cdot (1 + 0.25)^{10} \][/tex]
### Step 4: Calculate the Result
Now, we will calculate the value step-by-step:
1. Calculate [tex]\( 1 + r \)[/tex]:
[tex]\[ 1 + 0.25 = 1.25 \][/tex]
2. Raise 1.25 to the power of 10:
[tex]\[ 1.25^{10} \][/tex]
Without a calculator, the exact computation would be challenging. But typically, this value is:
[tex]\[ 1.25^{10} \approx 9.313225746 \][/tex]
3. Multiply this result by the initial number of infected people:
[tex]\[ 400 \cdot 9.313225746 \approx 3725.29 \][/tex]
### Step 5: Final Answer
Since we are dealing with people, we should round to the nearest whole number:
[tex]\[ \boxed{3725} \][/tex]
So, after 10 days, approximately 3725 people will be infected with the virus.
### Step 1: Understanding the Problem
We are given:
- The initial number of infected people, [tex]\( N_0 = 400 \)[/tex].
- The growth rate of the infection per day, [tex]\( r = 25\% = 0.25 \)[/tex].
- The duration to predict the number of infections, [tex]\( t = 10 \)[/tex] days.
We need to find how many people will be infected after 10 days.
### Step 2: Writing the Exponential Growth Function
The situation describes exponential growth, where the number of infected people grows by a fixed percentage each day. The general form of an exponential growth function is:
[tex]\[ N(t) = N_0 \cdot (1 + r)^t \][/tex]
Where:
- [tex]\( N(t) \)[/tex] is the number of infected people at time [tex]\( t \)[/tex].
- [tex]\( N_0 \)[/tex] is the initial number of infected people.
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time period (in days in this case).
### Step 3: Applying the Values to the Function
Let's apply the given values to the exponential growth function:
[tex]\[ N(10) = 400 \cdot (1 + 0.25)^{10} \][/tex]
### Step 4: Calculate the Result
Now, we will calculate the value step-by-step:
1. Calculate [tex]\( 1 + r \)[/tex]:
[tex]\[ 1 + 0.25 = 1.25 \][/tex]
2. Raise 1.25 to the power of 10:
[tex]\[ 1.25^{10} \][/tex]
Without a calculator, the exact computation would be challenging. But typically, this value is:
[tex]\[ 1.25^{10} \approx 9.313225746 \][/tex]
3. Multiply this result by the initial number of infected people:
[tex]\[ 400 \cdot 9.313225746 \approx 3725.29 \][/tex]
### Step 5: Final Answer
Since we are dealing with people, we should round to the nearest whole number:
[tex]\[ \boxed{3725} \][/tex]
So, after 10 days, approximately 3725 people will be infected with the virus.
