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.

Weeks Water Level
0 60
1 56.4
2 53.02
3 49.84
4 46.84

Answer 1

To model the water level of the lake over time with an exponential function, we can express the water level \( L(t) \) after \( t \) weeks, where the initial water level is \( 60 \) feet and it drops by \( 6\% \) each week.

First, let's convert the percentage drop into a decimal for the function. A \( 6\% \) drop means that \( 94\% \) of the water level remains after each week. This can be expressed as:

\[
0.94 = 1 - 0.06
\]

Now, we can write the exponential decay function. The general form of an exponential function for this scenario is:

\[
L(t) = L_0 \cdot (1 - r)^t
\]

Where:
- \( L_0 \) is the initial water level
- \( r \) is the rate of decrease (as a decimal)
- \( t \) is the time in weeks

Plugging in the values we have:

- \( L_0 = 60 \) (the initial water level)
- \( r = 0.06 \) (the rate of decrease)

Thus, the function becomes:

\[
L(t) = 60 \cdot (0.94)^t
\]

This function describes the water level in feet, \( L(t) \), after \( t \) weeks.

To summarize:

- **Exponential Function**: \( L(t) = 60 \cdot (0.94)^t \)

This function can now be used to calculate the water level for any week \( t \) during the drought.