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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. We are given that the initial area of moss is 11 square centimeters. 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 $A_n = A_0 \times r^n$, where:
- $A_n$ is the area after $n$ months,
- $A_0$ is the initial area,
- $r$ is the growth rate,
- $n$ is the number of months.
3. Identifying the Values
In our case, we have:
- $A_0 = 11$ square centimeters,
- $r = 1.5$,
- $n = 6$ months.
4. Substituting the Values
Plugging these values into the formula, we get:
$A_6 = 11 Imes (1.5)^6$
5. Calculating the Area
Now, we calculate $(1.5)^6$:
$(1.5)^6 = 11.390625$
Then, we multiply this by 11:
$A_6 = 11 Imes 11.390625 = 125.296875$
6. Finding the Answer
The area after 6 months is approximately 125.3 square centimeters. Comparing this with the given options, we see that option C is the closest.
### Examples
Exponential growth is a mathematical concept that appears frequently in nature and finance. For example, it can model the growth of a population, the accumulation of interest in a bank account, or the spread of a disease. Understanding exponential growth helps us make predictions and informed decisions in various real-world scenarios, such as planning for future resource needs or assessing the impact of an investment.
- 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. We are given that the initial area of moss is 11 square centimeters. 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 $A_n = A_0 \times r^n$, where:
- $A_n$ is the area after $n$ months,
- $A_0$ is the initial area,
- $r$ is the growth rate,
- $n$ is the number of months.
3. Identifying the Values
In our case, we have:
- $A_0 = 11$ square centimeters,
- $r = 1.5$,
- $n = 6$ months.
4. Substituting the Values
Plugging these values into the formula, we get:
$A_6 = 11 Imes (1.5)^6$
5. Calculating the Area
Now, we calculate $(1.5)^6$:
$(1.5)^6 = 11.390625$
Then, we multiply this by 11:
$A_6 = 11 Imes 11.390625 = 125.296875$
6. Finding the Answer
The area after 6 months is approximately 125.3 square centimeters. Comparing this with the given options, we see that option C is the closest.
### Examples
Exponential growth is a mathematical concept that appears frequently in nature and finance. For example, it can model the growth of a population, the accumulation of interest in a bank account, or the spread of a disease. Understanding exponential growth helps us make predictions and informed decisions in various real-world scenarios, such as planning for future resource needs or assessing the impact of an investment.
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