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.
a. The rate of change (slope) for this scenario can be calculated as the change in the water level divided by the change in the number of days:
Slope = (Change in water level) / (Change in days)
Slope = (62 - 34) / (10 - 3) = 28 / 7 = 4
b. The y-intercept for this scenario is the initial water level before the rain began, which is 22 feet.
c. To write the equation in slope-intercept form, we can use the slope (4) and the y-intercept (22):
y = mx + b
where:
y = water level
m = slope
x = number of days
b = y-intercept
Therefore, the equation in slope-intercept form is:
y = 4x + 22
d. To write this equation in standard form, we can rearrange the terms:
y = 4x + 22
Move 4x to the other side:
4x - y = -22
Multiply through by -1 to get the coefficient of x positive:
-y + 4x = -22
So, the equation in standard form is:
4x - y = -22
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 water levels (62 - 22 = 40 feet), and the change in x is the difference in days (10 - 0 = 10 days).
So, the slope is 40/10 = 4 feet per day.
b. The y-intercept is the initial water level before the rain began, which is given as 22 feet.
c. To write an equation for the scenario in slope-intercept form (y = mx + b), we can substitute the slope and y-intercept into the equation:
y = 4x + 22
d. To write this equation in standard form (Ax + By = C), we can rearrange the equation:
4x - y = -22
So, the equation in standard form is 4x - y = -22.
a. To find the rate of change (slope) in this scenario, we need to determine the change in the water level (y) divided by the change in the number of days (x).
Change in water level = Final water level - Initial water level = 62 ft - 22 ft = 40 ft
Change in number of days = Final number of days - Initial number of days = 10 days - 3 days = 7 days
Slope = Change in water level / Change in number of days = 40 ft / 7 days = 5.71 ft/day
b. The y-intercept represents the starting point or the water level before the rain began. In this scenario, the river was 22 feet before the rain started, so the y-intercept is 22.
c. The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept.
Using the values we calculated:
Slope (m) = 5.71 ft/day
Y-intercept (b) = 22 ft
Therefore, the equation in slope-intercept form is:
y = 5.71x + 22
d. To write the equation in standard form, we need to move the x term to the other side of the equation.
The standard form of an equation is Ax + By = C, where A, B, and C are constants.
In this case:
y = 5.71x + 22
To move the x term, we subtract 5.71x from both sides:
-5.71x + y = 22
Finally, to make the coefficient of x positive, we can multiply the entire equation by -1:
5.71x - y = -22
So, the equation in standard form is:
5.71x - y = -22