To answer the questions step by step, we start by analyzing the information given:
- Initial water level: 22 feet (before the rain began).
- Water level after 3 days of rain: 34 feet.
- Water level after 10 days of rain: 62 feet.
a. What is the rate of change (slope) for this scenario?
To find the slope (rate of change), we can use the formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points:
- \( (3, 34) \) for 3 days and 34 feet.
- \( (10, 62) \) for 10 days and 62 feet.
Applying the formula:
\[ \text{slope} = \frac{62 - 34}{10 - 3} = \frac{28}{7} = 4 \]
So the slope (rate of change) is 4.
b. What is the y-intercept for this scenario?
The y-intercept is the point where the line crosses the y-axis, which occurs when \( x = 0 \). We know that the river was at 22 feet before the rain began. Therefore, the y-intercept is the initial water level:
The y-intercept is 22.
c. Write an equation for the scenario in slope-intercept form.
The slope-intercept form of a line is given by:
\[ y = mx + b \]
Where \( m \) is the slope and \( b \) is the y-intercept. We already calculated \( m = 4 \) and \( b = 22 \).
Thus, the equation is:
\[ y = 4x + 22 \]
d. Write this equation in standard form.
To convert the slope-intercept form \( y = 4x + 22 \) into standard form \( Ax + By = C \):
- Rearranging gives us: \[ -4x + y = 22 \]
- To have A as a positive integer, multiply the entire equation by -1: \[ 4x - y = -22 \]
To convert it into a more standard form:
- Adding 22 to both sides to keep C positive: \[ 4x - y = 22 \]
So the equation in standard form is:
\[ 4x - y = 22 \]
Summary of Answers:
a. 4
b. 22
c. y = 4x + 22
d. 4x - y = 22
1 answer