Question

Day (x) 0 13 22
Depth in ft (y) 29.91 31.08 31.89
The water level of a lake is rising, and local residents have been keeping records in a data table. Use the table to derive an equation that describes the depth of the lake in feet, y, on each day, x. Then, use the equation to determine how many days will it take for the water level to reach 40 feet. Round to the nearest whole number.

(1 point)
It will take approximately days for the water level to reach 40 feet.

Answer 1

To derive the equation that describes the depth of the lake in feet, \(y\), on each day, \(x\), we can use the given data points to determine a linear equation in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

The given data points are:
1. (0, 29.91)
2. (13, 31.08)
3. (22, 31.89)

### Step 1: Calculate the slope (m)

We can calculate the slope using two points, for example, (0, 29.91) and (22, 31.89).

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{31.89 - 29.91}{22 - 0} = \frac{1.98}{22} \approx 0.09
\]

### Step 2: Write the equation

Using the slope of approximately 0.09 and the y-intercept from the first point (0, 29.91), we can write the equation as:

\[
y = 0.09x + 29.91
\]

### Step 3: Determine how many days it will take to reach 40 feet

To find out when the water level reaches 40 feet, we set \(y\) to 40 and solve for \(x\):

\[
40 = 0.09x + 29.91
\]

Subtract 29.91 from both sides:

\[
40 - 29.91 = 0.09x
\]

This gives:

\[
10.09 = 0.09x
\]

Now, divide both sides by 0.09:

\[
x = \frac{10.09}{0.09} \approx 112.11
\]

### Step 4: Round to the nearest whole number

Rounding 112.11 to the nearest whole number gives us approximately 112 days.

### Conclusion

It will take approximately **112 days** for the water level to reach 40 feet.