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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 $A_t = A_0
vert r^t$, with $A_0 = 11$, $r = 1.5$, and $t = 6$.
- Calculate the area after 6 months: $A_6 = 11
vert (1.5)^6 = 125.296875$.
- Round the result to one decimal place, which gives the final area as $\boxed{125.3 cm^2}$.
### 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 covered by moss multiplies by 1.5 each month. Paul returns in 6 months, and we need to find the approximate area of the moss after 6 months. This is an exponential growth problem.
2. Exponential Growth Formula
The formula for exponential growth is given by $A_t = A_0 \times r^t$, where:
- $A_t$ is the area after $t$ months,
- $A_0$ is the initial area,
- $r$ is the growth rate per month, and
- $t$ is the number of months.
3. Identifying the Values
In this problem, we have:
- $A_0 = 11$ square centimeters,
- $r = 1.5$, and
- $t = 6$ months.
4. Substituting the Values
Substituting these values into the formula, we get:
$A_6 = 11 \times (1.5)^6$
5. Calculating the Growth Factor
Now, we calculate $(1.5)^6$:
$(1.5)^6 = 1.5 \times 1.5 \times 1.5 \times 1.5 \times 1.5 \times 1.5 = 11.390625$
6. Calculating the Final Area
Next, we multiply this result by the initial area:
$A_6 = 11 \times 11.390625 = 125.296875$
7. Rounding the Result
Rounding this to one decimal place, we get $125.3$ square centimeters.
8. Selecting the Correct Answer
Comparing this with the given options, we see that the closest answer is $125.3 cm^2$.
### Examples
Exponential growth is a mathematical concept that describes the increase in a quantity over time. It can be used to model various real-world phenomena, such as 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 investments, planning for the future, and understanding the world around you.
- The formula for exponential growth is $A_t = A_0
vert r^t$, with $A_0 = 11$, $r = 1.5$, and $t = 6$.
- Calculate the area after 6 months: $A_6 = 11
vert (1.5)^6 = 125.296875$.
- Round the result to one decimal place, which gives the final area as $\boxed{125.3 cm^2}$.
### 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 covered by moss multiplies by 1.5 each month. Paul returns in 6 months, and we need to find the approximate area of the moss after 6 months. This is an exponential growth problem.
2. Exponential Growth Formula
The formula for exponential growth is given by $A_t = A_0 \times r^t$, where:
- $A_t$ is the area after $t$ months,
- $A_0$ is the initial area,
- $r$ is the growth rate per month, and
- $t$ is the number of months.
3. Identifying the Values
In this problem, we have:
- $A_0 = 11$ square centimeters,
- $r = 1.5$, and
- $t = 6$ months.
4. Substituting the Values
Substituting these values into the formula, we get:
$A_6 = 11 \times (1.5)^6$
5. Calculating the Growth Factor
Now, we calculate $(1.5)^6$:
$(1.5)^6 = 1.5 \times 1.5 \times 1.5 \times 1.5 \times 1.5 \times 1.5 = 11.390625$
6. Calculating the Final Area
Next, we multiply this result by the initial area:
$A_6 = 11 \times 11.390625 = 125.296875$
7. Rounding the Result
Rounding this to one decimal place, we get $125.3$ square centimeters.
8. Selecting the Correct Answer
Comparing this with the given options, we see that the closest answer is $125.3 cm^2$.
### Examples
Exponential growth is a mathematical concept that describes the increase in a quantity over time. It can be used to model various real-world phenomena, such as 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 investments, planning for the future, and understanding the world around you.
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