To analyze the water level change of the river over time, we can follow these steps:
a. What is the rate of change (slope) for this scenario?
We can find the slope using the two data points given:
- Point 1: \( (3, 45) \) — after 3 days, the water level is 45 feet.
- Point 2: \( (10, 10) \) — after 10 days, the water level is 10 feet.
The formula for the slope (rate of change) \( m \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting in the values:
\[
m = \frac{10 - 45}{10 - 3} = \frac{-35}{7} = -5
\]
So, the rate of change (slope) is:
-5
b. What is the y-intercept for this scenario?
The y-intercept is the value of \( y \) when \( x = 0 \). According to the problem, the water level was 60 feet before the drought began (at \( t = 0 \)).
So, the y-intercept is:
(0, 60)
c. Write an equation for the scenario in slope-intercept form.
Using the slope \( m = -5 \) and the y-intercept \( (0, 60) \), we can write the equation in slope-intercept form \( y = mx + b \):
\[
y = -5x + 60
\]
d. Write an equation in standard form.
To convert \( y = -5x + 60 \) to standard form \( Ax + By = C \), we can rearrange the equation:
- Add \( 5x \) to both sides:
\[
5x + y = 60
\]
So, the equation in standard form is:
5x + y = 60
Summary of Responses:
a. \(-5\)
b. \((0, 60)\)
c. \(y = -5x + 60\)
d. \(5x + y = 60\)
7 answers