- Identify the initial area as \(A_0 = 11 \text{ cm}^2\) and the growth rate per month as \(r = 1.5\).
- Apply the exponential growth formula \(A(t) = A_0 \cdot r^t\) for \(t = 6\) months.
- Calculate \(A(6) = 11 \times (1.5)^6 \approx 125.3 \text{ cm}^2\).
- Compare the result with the given options and select the correct value: \(\boxed{125.3 \text{ cm}^2}\).
Explanation
- Problem Analysis and Modeling
The moss growth is modeled by an exponential growth process. We are given the initial area of the moss, \(A_0 = 11 \text{ cm}^2\), and its growth rate per month, which is a factor of \(r = 1.5\). We want to find the area covered by the moss after \(t = 6\) months. The formula for exponential growth is given by \(A(t) = A_0 \cdot r^t\). In our case, this means we need to evaluate the expression \(A(6) = 11 \cdot (1.5)^6\).
- Step-by-Step Calculation
To calculate the area after 6 months, we first compute the growth factor over this time period: \((1.5)^6\).
Calculating \(1.5^6\):
\((1.5)^2 = 2.25\)
\((1.5)^3 = 3.375\)
\((1.5)^6 = (3.375)^2 \approx 11.390625\)
Next, we multiply this growth factor by the initial area \(A_0 = 11 \text{ cm}^2\):
\(A(6) = 11 \cdot 11.390625 = 125.296875 \text{ cm}^2\).
Rounding this value to one decimal place, we get approximately \(125.3 \text{ cm}^2\).
- Final Selection and Conclusion
Comparing our result of approximately \(125.3 \text{ cm}^2\) with the given options:
- \(16.5 \text{ cm}^2\)
- \(125.3 \text{ cm}^2\)
- \(14.7 \text{ cm}^2\)
- \(99.1 \text{ cm}^2\)
The calculated value matches the option \(125.3 \text{ cm}^2\). Thus, the correct area after 6 months is approximately \(\boxed{125.3 \text{ cm}^2}\).
Examples
Exponential growth models are used in various fields, such as biology, finance, and technology. For instance, if a bacterial colony doubles every hour, or if an investment earns compound interest, the quantity grows exponentially over time. Understanding this model allows us to project future values and make informed decisions, such as estimating population growth or calculating the future value of savings.