12. 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. Response area

d. Write this equation in standard form. Response area

Answer 1

To address the questions about the river water level after days of rain, let's first analyze the given information:

• Water level after 3 days of rain = 34 feet
• Water level after 10 days of rain = 62 feet
• Initial water level before the rain = 22 feet

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

To find the rate of change (slope), we can use the formula for slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where \((x_1, y_1)\) and \((x_2, y_2)\) are the two points:

• Point 1: (3, 34)
• Point 2: (10, 62)

Calculating the slope: \[ m = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4 \]

Response area: 4

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

The y-intercept is the initial water level before the rain began, which is given as 22 feet.

Response area: 22

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

The slope-intercept form of a linear equation is given by: \[ y = mx + b \] Where \(m\) is the slope and \(b\) is the y-intercept. Plugging in the values: \[ y = 4x + 22 \]

Response area: \(y = 4x + 22\)

d. Write this equation in standard form.

To convert the equation from slope-intercept form (\(y = mx + b\)) to standard form (\(Ax + By = C\)), we can rearrange it: \[ y - 4x = 22 \] Rearranging gives: \[ -4x + y = 22 \] To make A positive, we can multiply by -1: \[ 4x - y = -22 \]

Response area: \(4x - y = -22\)

In summary:

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