Question
The water level of a lake started at 60 feet. During a drought the water level dropped `6\%` each week. Write an exponential function that describes the water level in terms of the number of weeks.
Answers
GPT-4o mini
To express the water level of the lake in terms of the number of weeks \( t \), we can use an exponential decay function.
The initial water level is 60 feet, and it drops by 6% each week. This means that 94% of the water level remains after each week. The remaining percentage can be expressed as a decimal:
\[
1 - 0.06 = 0.94
\]
Now, we can write the exponential function \( W(t) \) for the water level \( W \) after \( t \) weeks. The formula for exponential decay is:
\[
W(t) = W_0 \cdot (r)^t
\]
where:
- \( W_0 \) is the initial amount (60 feet),
- \( r \) is the decay factor (0.94),
- \( t \) is the number of weeks.
Substituting these values into the formula, we have:
\[
W(t) = 60 \cdot (0.94)^t
\]
This function \( W(t) \) describes the water level of the lake in feet after \( t \) weeks.
The initial water level is 60 feet, and it drops by 6% each week. This means that 94% of the water level remains after each week. The remaining percentage can be expressed as a decimal:
\[
1 - 0.06 = 0.94
\]
Now, we can write the exponential function \( W(t) \) for the water level \( W \) after \( t \) weeks. The formula for exponential decay is:
\[
W(t) = W_0 \cdot (r)^t
\]
where:
- \( W_0 \) is the initial amount (60 feet),
- \( r \) is the decay factor (0.94),
- \( t \) is the number of weeks.
Substituting these values into the formula, we have:
\[
W(t) = 60 \cdot (0.94)^t
\]
This function \( W(t) \) describes the water level of the lake in feet after \( t \) weeks.