Question

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) 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

is d also y=4x-22

Answer 2

To solve the problem, we can first determine some key components using the given information.

### Given Information:
- Water level after 3 days of rain: \(34\) feet
- Water level after 10 days of rain: \(62\) feet
- Initial water level before rain began: \(22\) feet

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

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

The rate of change (slope) can be calculated using the formula:

\[
\text{slope (m)} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Where:
- \((x_1, y_1) = (3, 34)\)
- \((x_2, y_2) = (10, 62)\)

Now, substituting in the values:

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

**Response for a:** 4

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

The y-intercept can be determined using the initial water level when no rain has occurred (i.e., when \(x = 0\)). Given that the river was 22 feet before the rain began:

**Response for b:** 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
- \(b\) is the y-intercept

Substituting the values we found:

\[
y = 4x + 22
\]

**Response for c:** \(y = 4x + 22\)

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

The standard form of a linear equation is given by:

\[
Ax + By = C
\]

Starting with the slope-intercept form:

\[
y = 4x + 22
\]

We can rearrange this into standard form:

\[
-4x + y = 22
\]

Multiplying through by -1 to keep \(A\) non-negative:

\[
4x - y = -22
\]

To make the right side positive, we can also write:

\[
4x - y = 22
\]

**Response for d:** \(4x - y = 22\)

### Summary of Responses:
a. **4**

b. **22**

c. **y = 4x + 22**

d. **4x - y = 22**

Answer 3

Let's clarify the equations to ensure we have the correct forms.

The slope-intercept form derived earlier is:

\[
y = 4x + 22
\]

To convert it to standard form \(Ax + By = C\), we can rearrange it. However, your suggestion of \(y = 4x - 22\) doesn't align with the information given.

Let's break down the valid conversions properly:

1. **Starting from the correct slope-intercept form:**
\[
y = 4x + 22
\]

2. **Rearranging to standard form:**
Subtract \(4x\) from both sides to get:
\[
-4x + y = 22
\]

To express it typically with \(A\) being positive, we multiply the entire equation by -1:
\[
4x - y = -22
\]

This equation can also be expressed as:

\[
4x - y = 22
\]

In conclusion:

- The correct version of standard form based on our slope-intercept equation \(y = 4x + 22\) remains:
\[
4x - y = 22
\]

Thus, **d. is not \(y = 4x - 22\)**; instead, the correct standard form is indeed:

\[
4x - y = 22
\]

Thank you for pointing that out, and I hope this provides clarity!