Question
Currently, 1,200 square meters of Pineview Lake is covered by algae. Scientists just added beneficial bacteria to the water, and they expect the area of the lake covered by algae to decrease by 10% each week. You can use a function to describe how much of the lake will still be covered by algae x weeks from now. Is the function linear or exponential? Which equation represents the function?
Answers
GPT-4o mini
The situation described involves a decrease in the area covered by algae by a percentage each week, which indicates that the function is **exponential** rather than linear. In linear functions, a constant amount is added or subtracted over equal intervals, while in exponential functions, a constant percentage of the value is reduced over equal intervals.
To represent the area of Pineview Lake covered by algae as a function of time, we can use the following exponential decay formula:
\[
A(x) = A_0 \cdot (1 - r)^x
\]
Where:
- \( A(x) \) is the area covered by algae after \( x \) weeks.
- \( A_0 \) is the initial area covered by algae (1,200 square meters).
- \( r \) is the rate of decay (10%, or 0.10).
- \( x \) is the number of weeks.
Substituting the values into the formula, we have:
\[
A(x) = 1200 \cdot (1 - 0.10)^x
\]
Simplifying further:
\[
A(x) = 1200 \cdot (0.90)^x
\]
This equation describes the area of the lake that will still be covered by algae \( x \) weeks from now. Since the area decreases by a percentage each week, the function is indeed **exponential**.
To represent the area of Pineview Lake covered by algae as a function of time, we can use the following exponential decay formula:
\[
A(x) = A_0 \cdot (1 - r)^x
\]
Where:
- \( A(x) \) is the area covered by algae after \( x \) weeks.
- \( A_0 \) is the initial area covered by algae (1,200 square meters).
- \( r \) is the rate of decay (10%, or 0.10).
- \( x \) is the number of weeks.
Substituting the values into the formula, we have:
\[
A(x) = 1200 \cdot (1 - 0.10)^x
\]
Simplifying further:
\[
A(x) = 1200 \cdot (0.90)^x
\]
This equation describes the area of the lake that will still be covered by algae \( x \) weeks from now. Since the area decreases by a percentage each week, the function is indeed **exponential**.