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

Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
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

Let's solve this step by step based on the information given:

1. Rate of Change (Slope):

   • The slope can be calculated using the formula: \[ \text{slope} (m) = \frac{y_2 - y_1}{x_2 - x_1} \]
   • Here, \(y_1 = 34\) feet (water level after 3 days), \(y_2 = 62\) feet (water level after 10 days), \(x_1 = 3\), and \(x_2 = 10\).
   • Calculating the slope: \[ m = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4 \]
   • Response for a: 4

2. Y-Intercept:

   • The y-intercept is the water level before the rain began. The problem states that the river was 22 feet before it started raining.
   • Response for b: 22

3. Equation in Slope-Intercept Form (y = mx + b):

   • We have the slope \(m = 4\) and the y-intercept \(b = 22\).
   • The equation in slope-intercept form is: \[ y = 4x + 22 \]
   • Response for c: y = 4x + 22

4. Equation in Standard Form:

   • To convert from slope-intercept form \(y = 4x + 22\) to standard form \(Ax + By = C\), we rearrange the equation: \[ -4x + y = 22 \]
   • To have A be a positive integer, we can multiply the entire equation by -1: \[ 4x - y = -22 \]
   • Response for d: 4x - y = -22

So, the finalized responses are: a. 4
b. 22
c. y = 4x + 22
d. 4x - y = -22