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)

Question

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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.
d. Write this equation in standard form.

Answer 1

To solve this problem, we will find the rate of change (slope) and the y-intercept for the given scenario involving the river's water level after a certain number of days of rain.

Let \( x \) represent the number of days of rain and \( y \) represent the water level of the river in feet.

Given:

• After 3 days of rain (\( x = 3 \)), the water level is 34 feet (\( y = 34 \)).
• After 10 days of rain (\( x = 10 \)), the water level is 62 feet (\( y = 62 \)).
• Before the rain began, the water level was 22 feet.

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

To find the slope (\( m \)), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \( (3, 34) \) and \( (10, 62) \):

• \( y_1 = 34 \), \( x_1 = 3 \)
• \( y_2 = 62 \), \( x_2 = 10 \) \[ m = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4 \]

Response for a: 4

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

The y-intercept is the value of \( y \) when \( x = 0 \). We can use the slope-intercept form of the equation, which is: \[ y = mx + b \] We know \( m = 4 \), and we need to find \( b \). We can use one of the points to solve for \( b \). Using the point \( (3, 34) \): \[ 34 = 4 \cdot 3 + b \ 34 = 12 + b \ b = 34 - 12 = 22 \]

Response for b: 22

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

Using the slope and y-intercept we found: \[ y = 4x + 22 \]

Response for c: y = 4x + 22

d. Write this equation in standard form.

To convert \( y = 4x + 22 \) to standard form \( Ax + By = C \):

1. Rearrange the equation: \[ -4x + y = 22 \]
2. To have \( A \) as a positive integer, multiply through by -1: \[ 4x - y = -22 \]

Response for d: 4x - y = -22

In summary:

• a: 4
• b: 22
• c: y = 4x + 22
• d: 4x - y = -22