Question

Suppose that the water level of a river is 34 feet after 3 days of rain and 62 feet after 10 days of rain. The river was 22 feet before the rain began.(4 points)
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a. What is the rate of change (slope) for this scenario? Response area

b. What is the y-intercept for this scenario? Response area

c. Write an equation for the scenario in slope-intercept form. Response area

d. Write this equation in standard form. Response area

Answer 1

To analyze the given scenario, we will define our variables first. Let \( x \) represent the number of days since the rain began, and let \( y \) represent the water level of the river in feet. Based on the information provided:

- After 3 days (when \( x = 3 \)), the water level \( y = 34 \) feet.
- After 10 days (when \( x = 10 \)), the water level \( y = 62 \) feet.
- The initial water level before it rained was \( 22 \) feet (when \( x = 0 \)).

### a. What is the rate of change (slope) for this scenario?

The slope \( m \) can be calculated by using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Taking the values from the points (3, 34) and (10, 62):
\[
m = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4
\]

**Response: 4**

### b. What is the y-intercept for this scenario?

The y-intercept can be found from the initial condition, which states that the water level was 22 feet before it rained, i.e., when \( x = 0 \), \( y = 22 \).

**Response: 22**

### c. Write an equation for the scenario in slope-intercept form.

The slope-intercept form of a linear equation is given by \( y = mx + b \). We have the slope \( m = 4 \) and the y-intercept \( b = 22 \).

So, the equation becomes:
\[
y = 4x + 22
\]

**Response: y = 4x + 22**

### d. Write this equation in standard form.

To convert the slope-intercept form \( y = 4x + 22 \) to standard form \( Ax + By = C \), we can rearrange it:
\[
4x - y = -22
\]

To express it in the standard form where \( A, B, C \) are integers, we can multiply all terms by -1:
\[
-4x + y = 22 \quad \text{(or rearranging the signs gives)} \quad 4x - y = -22
\]

**Response: 4x - y = -22**

Note: The standard form can have different appearances depending on how it is expressed, but as long as the equation is a linear equation in \( Ax + By = C \) form with integer coefficients, it is acceptable.