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)
a. What is the rate of change (slope) for this scenario?
What is the y-intercept for this scenario?
c. Write an equation for the scenario in slope-intercept form
. Write this equation in standard form.
To find the rate of change (slope), we can use the formula:
slope = (change in y) / (change in x)
The change in y is the difference in water levels, which is 62 - 34 = 28 feet.
The change in x is the difference in days, which is 10 - 3 = 7 days.
So, the slope is: slope = 28 / 7 = 4 feet per day.
Since the river was 22 feet before the rain began, the y-intercept is 22.
Therefore, the equation for the scenario in slope-intercept form is:
y = 4x + 22
To write it in standard form, we need to rearrange the equation:
-4x + y = 22
To find the rate of change (slope) for this scenario, we can use the formula for slope:
Slope = (Change in y) / (Change in x)
In this case, the change in y is the difference in the water level of the river, which is 62 - 22 = 40 feet. The change in x is the number of days, which is 10 - 0 = 10 days.
Slope = (40 feet) / (10 days) = 4 feet per day
So, the slope for this scenario is 4 feet per day.
To find the y-intercept, we can use the initial water level of the river, which is 22 feet. The y-intercept represents the starting value when x = 0.
Therefore, the y-intercept for this scenario is 22 feet.
Now, let's write an equation for the scenario in slope-intercept form, which is given by:
y = mx + b
Where:
y is the water level of the river
m is the slope
x is the number of days
b is the y-intercept
Substituting in the values we found:
y = 4x + 22
This is the equation for the scenario in slope-intercept form.
To write this equation in standard form, we need to rearrange the equation to have all the variables on one side and the constant term on the other side:
-4x + y = 22
This is the equation for the scenario in standard form.
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, which is 62 - 22 = 40 feet. The change in x is the difference in days, which is 10 - 0 = 10 days (since we are comparing the initial level to after 10 days of rain).
slope = (40 feet) / (10 days)
Now we can simplify:
slope = 4 feet per day
So the rate of change (slope) for this scenario is 4 feet per day.
To find the y-intercept, we need to determine the initial water level when there were 0 days of rain. From the given information, the river was 22 feet before the rain began. Therefore, the y-intercept is 22 feet.
Now let's write the equation for the scenario in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
The slope (m) is 4 feet per day, and the y-intercept (b) is 22 feet. Plugging these values into the equation, we get:
y = 4x + 22
Now, let's write this equation in standard form, which is Ax + By = C.
Rearranging the equation, we have:
-4x + 1y = -22
Multiplying through by -1 to make the coefficient of x positive, we get:
4x - y = 22
Therefore, the equation for the scenario in slope-intercept form is y = 4x + 22, and in standard form, it is 4x - y = 22.