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Answer :
- The problem describes exponential growth of moss, where the area multiplies by 1.5 each month.
- The formula for exponential growth is $A_n = A_0 Imes r^n$.
- Substitute the given values: $A_0 = 11$, $r = 1.5$, and $n = 6$.
- Calculate the area after 6 months: $A_6 = 11 Imes (1.5)^6 = 125.3$ cm$^2$. The final answer is $\boxed{125.3 cm^2}$.
### Explanation
1. Understanding the Problem
Let's analyze the problem. Paul measured an initial area of moss, and we know that the area grows by a factor of 1.5 each month. We need to find the area after 6 months. This is an exponential growth problem.
2. Stating the Formula
The formula for exponential growth is given by:
$$A_n = A_0 Imes r^n$$
where:
- $A_n$ is the area after $n$ months,
- $A_0$ is the initial area,
- $r$ is the growth factor,
- $n$ is the number of months.
3. Identifying the Values
In this case, we have:
- $A_0 = 11$ cm$^2$
- $r = 1.5$
- $n = 6$ months
So, we need to calculate $A_6$.
4. Substituting the Values
Substituting the values into the formula, we get:
$$A_6 = 11 Imes (1.5)^6$$
5. Calculating the Area
Calculating $(1.5)^6$, we get:
$$(1.5)^6 = 11.390625$$
Then, multiplying by 11:
$$A_6 = 11 Imes 11.390625 = 125.296875$$
6. Finding the Answer
The area after 6 months is approximately 125.3 cm$^2$. Comparing this with the given options, we see that option B is the closest.
### Examples
Exponential growth is a common phenomenon in nature and finance. For instance, the growth of bacteria in a culture, the increase in population over time, or the accumulation of interest in a savings account can all be modeled using exponential functions. Understanding exponential growth helps us predict future values based on current trends, which is crucial for making informed decisions in various fields.
- The formula for exponential growth is $A_n = A_0 Imes r^n$.
- Substitute the given values: $A_0 = 11$, $r = 1.5$, and $n = 6$.
- Calculate the area after 6 months: $A_6 = 11 Imes (1.5)^6 = 125.3$ cm$^2$. The final answer is $\boxed{125.3 cm^2}$.
### Explanation
1. Understanding the Problem
Let's analyze the problem. Paul measured an initial area of moss, and we know that the area grows by a factor of 1.5 each month. We need to find the area after 6 months. This is an exponential growth problem.
2. Stating the Formula
The formula for exponential growth is given by:
$$A_n = A_0 Imes r^n$$
where:
- $A_n$ is the area after $n$ months,
- $A_0$ is the initial area,
- $r$ is the growth factor,
- $n$ is the number of months.
3. Identifying the Values
In this case, we have:
- $A_0 = 11$ cm$^2$
- $r = 1.5$
- $n = 6$ months
So, we need to calculate $A_6$.
4. Substituting the Values
Substituting the values into the formula, we get:
$$A_6 = 11 Imes (1.5)^6$$
5. Calculating the Area
Calculating $(1.5)^6$, we get:
$$(1.5)^6 = 11.390625$$
Then, multiplying by 11:
$$A_6 = 11 Imes 11.390625 = 125.296875$$
6. Finding the Answer
The area after 6 months is approximately 125.3 cm$^2$. Comparing this with the given options, we see that option B is the closest.
### Examples
Exponential growth is a common phenomenon in nature and finance. For instance, the growth of bacteria in a culture, the increase in population over time, or the accumulation of interest in a savings account can all be modeled using exponential functions. Understanding exponential growth helps us predict future values based on current trends, which is crucial for making informed decisions in various fields.
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