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Answer :
- The problem describes exponential growth of moss, where the area multiplies by 1.5 each month.
- Apply the exponential growth formula: $A = P(1 + r)^t$, where $P = 11$, $r = 0.5$, and $t = 6$.
- Calculate the area after 6 months: $A = 11 \,\text{cm}^2 \times (1.5)^6 = 125.296875 \,\text{cm}^2$.
- Round the result to one decimal place, resulting in the final area: $\boxed{125.3 \,\text{cm}^2}$.
### 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:
$$A = P(1 + r)^t$$
where:
- $A$ is the final amount
- $P$ is the initial amount
- $r$ is the growth rate (as a decimal)
- $t$ is the time period
3. Identifying the Values
In this case, we have:
- $P = 11$ square centimeters
- The area multiplies by 1.5 each month, so the growth factor is 1.5. This means $1 + r = 1.5$, so $r = 0.5$.
- $t = 6$ months
4. Substituting the Values
Now, we substitute these values into the formula:
$$A = 11 \times (1.5)^6$$
5. Calculating the Final Area
Calculating $(1.5)^6$:
$$(1.5)^6 = 11.390625$$
Now, multiply this by 11:
$$A = 11 \,\text{cm}^2 \times 11.390625 = 125.296875 \,\text{cm}^2$$
Rounding to one decimal place, we get:
$$A \approx 125.3 \,\text{cm}^2$$
6. Final Answer
Therefore, the approximate area of the moss after 6 months will be $125.3 \,\text{cm}^2$.
### Examples
Exponential growth is a mathematical concept that describes the increase in a quantity over time. It has many real-world applications, such as calculating population growth, compound interest, and the spread of diseases. For example, if you invest money in a savings account with compound interest, the amount of money you have will grow exponentially over time. Similarly, the population of a city or country can grow exponentially if the birth rate is higher than the death rate. Understanding exponential growth can help you make informed decisions about your finances, health, and other aspects of your life.
- Apply the exponential growth formula: $A = P(1 + r)^t$, where $P = 11$, $r = 0.5$, and $t = 6$.
- Calculate the area after 6 months: $A = 11 \,\text{cm}^2 \times (1.5)^6 = 125.296875 \,\text{cm}^2$.
- Round the result to one decimal place, resulting in the final area: $\boxed{125.3 \,\text{cm}^2}$.
### 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:
$$A = P(1 + r)^t$$
where:
- $A$ is the final amount
- $P$ is the initial amount
- $r$ is the growth rate (as a decimal)
- $t$ is the time period
3. Identifying the Values
In this case, we have:
- $P = 11$ square centimeters
- The area multiplies by 1.5 each month, so the growth factor is 1.5. This means $1 + r = 1.5$, so $r = 0.5$.
- $t = 6$ months
4. Substituting the Values
Now, we substitute these values into the formula:
$$A = 11 \times (1.5)^6$$
5. Calculating the Final Area
Calculating $(1.5)^6$:
$$(1.5)^6 = 11.390625$$
Now, multiply this by 11:
$$A = 11 \,\text{cm}^2 \times 11.390625 = 125.296875 \,\text{cm}^2$$
Rounding to one decimal place, we get:
$$A \approx 125.3 \,\text{cm}^2$$
6. Final Answer
Therefore, the approximate area of the moss after 6 months will be $125.3 \,\text{cm}^2$.
### Examples
Exponential growth is a mathematical concept that describes the increase in a quantity over time. It has many real-world applications, such as calculating population growth, compound interest, and the spread of diseases. For example, if you invest money in a savings account with compound interest, the amount of money you have will grow exponentially over time. Similarly, the population of a city or country can grow exponentially if the birth rate is higher than the death rate. Understanding exponential growth can help you make informed decisions about your finances, health, and other aspects of your life.
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