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Answer :
- The problem involves exponential growth of moss, where the area multiplies by 1.5 each month.
- Calculate the growth factor after 6 months: $(1.5)^6 = 11.390625$.
- Multiply the initial area by the growth factor: $11
11.390625 = 125.296875$.
- The approximate area of moss after 6 months is $\boxed{125.3 cm^2}$.
### Explanation
1. Problem Analysis
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. Exponential Growth Formula
The formula for exponential growth is $A_n = A_0
^n$, where $A_n$ is the area after $n$ months, $A_0$ is the initial area, and $r$ is the growth rate. In this case, $A_0 = 11$, $r = 1.5$, and $n = 6$.
3. Calculating the Growth Factor
Now, we plug in the values into the formula: $A_6 = 11
(1.5)^6$. We need to calculate $(1.5)^6$.
$(1.5)^2 = 2.25$
$(1.5)^3 = 2.25
1.5 = 3.375$
$(1.5)^4 = 3.375
1.5 = 5.0625$
$(1.5)^5 = 5.0625
1.5 = 7.59375$
$(1.5)^6 = 7.59375
1.5 = 11.390625$
4. Calculating the Final Area
Now we multiply this by the initial area: $A_6 = 11
11.390625 = 125.296875$.
5. Selecting the Correct Answer
We need to choose the closest answer from the given options: A. $16.5 cm^2$, B. $14.7 cm^2$, C. $99.1 cm^2$, D. $125.3 cm^2$. The closest value to $125.296875$ is $125.3 cm^2$.
6. Final Answer
Therefore, the approximate area of the moss after 6 months is $125.3 cm^2$.
### Examples
Exponential growth is a common phenomenon in nature and finance. For example, the growth of bacteria in a culture, the increase in population over time, and the accumulation of compound interest in a savings account all follow exponential growth patterns. Understanding exponential growth helps us predict future outcomes and make informed decisions.
- Calculate the growth factor after 6 months: $(1.5)^6 = 11.390625$.
- Multiply the initial area by the growth factor: $11
11.390625 = 125.296875$.
- The approximate area of moss after 6 months is $\boxed{125.3 cm^2}$.
### Explanation
1. Problem Analysis
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. Exponential Growth Formula
The formula for exponential growth is $A_n = A_0
^n$, where $A_n$ is the area after $n$ months, $A_0$ is the initial area, and $r$ is the growth rate. In this case, $A_0 = 11$, $r = 1.5$, and $n = 6$.
3. Calculating the Growth Factor
Now, we plug in the values into the formula: $A_6 = 11
(1.5)^6$. We need to calculate $(1.5)^6$.
$(1.5)^2 = 2.25$
$(1.5)^3 = 2.25
1.5 = 3.375$
$(1.5)^4 = 3.375
1.5 = 5.0625$
$(1.5)^5 = 5.0625
1.5 = 7.59375$
$(1.5)^6 = 7.59375
1.5 = 11.390625$
4. Calculating the Final Area
Now we multiply this by the initial area: $A_6 = 11
11.390625 = 125.296875$.
5. Selecting the Correct Answer
We need to choose the closest answer from the given options: A. $16.5 cm^2$, B. $14.7 cm^2$, C. $99.1 cm^2$, D. $125.3 cm^2$. The closest value to $125.296875$ is $125.3 cm^2$.
6. Final Answer
Therefore, the approximate area of the moss after 6 months is $125.3 cm^2$.
### Examples
Exponential growth is a common phenomenon in nature and finance. For example, the growth of bacteria in a culture, the increase in population over time, and the accumulation of compound interest in a savings account all follow exponential growth patterns. Understanding exponential growth helps us predict future outcomes and make informed decisions.
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