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.

Answer 1

To solve the problem, we first need to organize the data given and use it to calculate the slope, y-intercept, and the equations.

Given data:

• The river level after 3 days of rain: \(34\) feet
• The river level after 10 days of rain: \(62\) feet
• The river level before the rain began (at \(0\) days): \(22\) feet

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

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

The slope (m) can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, using the points \((3, 34)\) and \((10, 62)\):

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

So, the slope is 4.

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

The y-intercept occurs when \(x = 0\). We were given that the river level before the rain began was \(22\) feet. Therefore, the y-intercept (b) is 22.

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

The slope-intercept form of a line is given by:

\[ y = mx + b \]

Substituting the values of \(m\) (slope) and \(b\) (y-intercept):

\[ y = 4x + 22 \]

d. Write this equation in standard form.

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

\[ -4x + y = 22 \]

To follow the standard form convention (where \(A\) is positive), we can multiply through by -1:

\[ 4x - y = -22 \]

So the standard form is:

\[ 4x - y = -22 \]

Summary of Responses

a. 4

b. 22

c. y = 4x + 22

d. 4x - y = -22