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: 0, 1, 2, 3, 4.
Water Level: 60, 56.4, 53.02, 49.84, 46.84.

y=_____

Answer 1

To describe the water level of the lake as a function of the number of weeks, we can use an exponential decay function.

Given that the initial water level (at week 0) is 60 feet and that it decreases by 6% each week, we can find the exponential function as follows:

1. The initial amount (at \(t = 0\)) is \(y_0 = 60\).
2. The decay rate is 6%, which means that 94% of the water level remains each week (100% - 6% = 94% = 0.94).

The general form of an exponential decay function can be written as:

\[
y = y_0 \cdot (1 - r)^t
\]

Where:
- \(y\) is the water level after \(t\) weeks
- \(y_0\) is the initial water level (60 feet)
- \(r\) is the rate of decrease (0.06 in this case)
- \(t\) is the number of weeks

Substituting the values into the equation:

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

So, the exponential function that describes the water level in terms of the number of weeks \(t\) is:

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