11. 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)
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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. Response area
a. The rate of change (slope) for this scenario can be calculated by finding the difference in water level (change in y) divided by the difference in time (change in x).
The change in water level is 62 - 34 = 28 feet.
The change in time is 10 - 3 = 7 days.
So the slope is 28 feet / 7 days = 4 feet per day.
b. The y-intercept for this scenario is the starting water level before the rain began, which is 22 feet.
c. To write the equation in slope-intercept form, we can use the slope (m) and the y-intercept (b).
The slope is 4 feet per day, and the y-intercept is 22 feet.
So the equation in slope-intercept form is y = 4x + 22.
d. To write the equation in standard form, we need to rearrange the equation to have the x and y terms on the same side and the constant term on the other side.
Starting with y = 4x + 22, we can subtract 4x from both sides to get -4x + y = 22.
So the equation in standard form is -4x + y = 22.
a. To find the rate of change (slope) for this scenario, we need to calculate the change in the water level divided by the change in the number of days.
Change in water level = 62 feet - 22 feet = 40 feet
Change in number of days = 10 days - 3 days = 7 days
Rate of change (slope) = Change in water level / Change in number of days
Rate of change (slope) = 40 feet / 7 days = 5.714 feet/day
b. The y-intercept for this scenario is the initial water level before the rain began, which is 22 feet.
c. To write an equation for the scenario in slope-intercept form (y = mx + b), we can use the rate of change (slope) and the y-intercept.
Equation: y = 5.714x + 22
d. To write this equation in standard form (Ax + By = C), we need to move all the terms to one side of the equation and have the coefficient of x positive.
Equation: -5.714x + 1y = -79.998
(Note: The decimals in the equation have been rounded for simplicity.)
a. To find the rate of change (slope) for this scenario, we can use the formula:
Slope = (change in y) / (change in x)
In this case, the change in y is the difference in the water level after 10 days and before the rain began, which is 62 - 22 = 40 feet. The change in x is the number of days, which is 10 - 0 = 10.
So, the slope is 40 / 10 = 4 feet per day.
b. The y-intercept is the value of y when x = 0, in other words, it is the initial water level before the rain began. In this scenario, the water level was 22 feet before the rain began, so the y-intercept is 22.
c. To write an equation for the scenario in slope-intercept form (y = mx + b), we can substitute the values we obtained into the equation:
y = 4x + 22
d. To write this equation in standard form (Ax + By = C), we need to rearrange the equation:
-4x + y = 22
Therefore, the answers are:
a. The rate of change (slope) for this scenario is 4 feet per day.
b. The y-intercept for this scenario is 22.
c. The equation for the scenario in slope-intercept form is y = 4x + 22.
d. The equation in standard form is -4x + y = 22.