To solve the problem, let's analyze the information provided step by step.
a. Rate of Change (Slope)
The slope can be calculated using the formula for slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here:
- \(y_1 = 34\) feet (after 3 days)
- \(y_2 = 62\) feet (after 10 days)
- \(x_1 = 3\) days
- \(x_2 = 10\) days
Calculating the slope: \[ m = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4 \]
Answer a: The rate of change (slope) for this scenario is 4.
b. Y-Intercept
Now we will find the y-intercept. The y-intercept is the starting value (when \(x = 0\)). Before the rain began, the river was at a level of 22 feet.
Answer b: The y-intercept for this scenario is 22.
c. Equation in Slope-Intercept Form
The general form of the slope-intercept equation is: \[ y = mx + b \] Where:
- \(m\) is the slope (which we found to be 4)
- \(b\) is the y-intercept (which we found to be 22)
Thus, the equation becomes: \[ y = 4x + 22 \]
Answer c: The equation for the scenario in slope-intercept form is \(y = 4x + 22\).
d. Equation in Standard Form
To write the equation \(y = 4x + 22\) in standard form \(Ax + By = C\), we rearrange it: \[ -4x + y = 22 \] Alternatively, we can multiply through by -1 for a different standard form: \[ 4x - y = -22 \] However, both versions are standard forms.
One of the standard forms using the option choices is \( -4x + y = 22\).
Answer d: The equation in standard form is \(-4x + y = 22\).
Summary of Answers:
- a: 4
- b: 22
- c: \(y = 4x + 22\)
- d: \(-4x + y = 22\)