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.

Question

GPT-4o mini · Answer

To determine the slope, y-intercept, and write the equations for this scenario, let's first summarize the information given:

The initial water level before the rain began is 22 feet.
After 3 days of rain, the water level is 34 feet.
After 10 days of rain, the water level is 62 feet.

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

We can find the slope ($m$) using the formula:

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

where $(x_1, y_1) = (3, 34)$ and $(x_2, y_2) = (10, 62)$.

Calculating the slope:

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

Response area: 4 (the rate of change is 4 feet per day)

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

The y-intercept is the value of $y$ when $x = 0$. From the problem statement, we know that before the rain began, the water level was 22 feet.

Response area: 22 (the y-intercept is 22 feet)

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

The slope-intercept form of a linear equation is given by:

\[ y = mx + b \]

Substituting the slope ($m = 4$) and y-intercept ($b = 22$) into the equation:

\[ y = 4x + 22 \]

Response area: $y = 4x + 22$

d. Write this equation in standard form.

The standard form of a linear equation is given by:

\[ Ax + By = C \]

Starting from the slope-intercept form $y = 4x + 22$, we can rearrange it into standard form:

Subtract $4x$ from both sides: \[ -4x + y = 22 \]
To get $A$ positive, we can multiply the entire equation by -1: \[ 4x - y = -22 \]

Response area: $4x - y = -22$

To summarize:

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

Answer

Write this equation in standard form.

GPT-4o mini · Answer

To rewrite the equation $y = 4x + 22$ in standard form, where the standard form is typically represented as $Ax + By = C$ with $A$, $B$, and $C$ being integers, and $A$ being non-negative, we follow these steps: 1. Start with the given equation: $$ y = 4x + 22 $$ 2. Rearrange it to move all terms involving $x$ and $y$ to one side: $$ -4x + y = 22 $$ 3. To write the equation in the traditional standard form where $A$ is positive, we can multiply the entire equation by $-1$: $$ 4x - y = -22 $$ So, the standard form of the equation is: **Response area**: $4x - y = -22$

Answer

answersy = 4x + 34 4x + y = 34 :: (0,0) 7 -4x + y = 22 28 -4x + y = 34 (0,4) y = 28x + 34 22 (0,34) y = 42-22

GPT-4o mini · Answer

It seems you have provided a series of equations and values that do not fully align with the context from the river scenario we discussed earlier. However, let’s match your input to various parts of the original question and clarify or correct where necessary, focusing only on standard form based on the rate of change calculated earlier.

From the river example:

Original slope-intercept equation (from previous analysis):
$ y = 4x + 22 $
Standard form (as derived earlier):
$ 4x - y = -22 $

Analysis of Provided Inputs:

$ y = 4x + 34 $: This does not match our previous findings.
$ 4x + y = 34 $: This does not match either.
Points like $ (0,0) $, $ (0,4) $, $ (0,34) $: These may refer to intercept points but are not needed here.
Values like "7", "28", "22": These seem arbitrary without context.
The equations like $ -4x + y = 22 $ and $ -4x + y = 34 $ are derived from different calculations and do not apply here.
The equation $ y = 28x + 34 $ appears incorrect in this context.

Correct Recap of Standard Form:

To restate clearly, based on the scenario given:

The standard form of the equation we've derived would be: \[ 4x - y = -22 \]

If you need any specific analysis or want clarification on any of the equations you've listed, please let me know!