Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. The river was 60 feet before the drought began.(4 points)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 an equation in standard form.
To find the rate of change (slope), we can use the formula:
slope = change in y / change in x
In this scenario, the change in y is the difference in the water level, which is 45 feet - 10 feet = 35 feet. The change in x is the difference in the number of days, which is 10 days - 3 days = 7 days.
So the slope is 35 feet / 7 days = 5 feet/day.
b. The y-intercept is the value of y when x is 0. In this scenario, at the beginning of the drought, the water level was 60 feet. Therefore, the y-intercept is 60.
c. To write the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we have:
y = 5x + 60
d. The standard form of an equation is written as Ax + By = C, where A, B, and C are constants. We can rearrange the slope-intercept form equation to standard form by moving all terms to one side:
-5x + y = 60
Multiplying through by -1 to make the coefficient of x positive:
5x - y = -60
Therefore, the equation in standard form is 5x - y = -60.
Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. Given the two points we know: (3, 45) and (10, 10), what would the point slope equation look like?
To find the point-slope equation using the two given points (3, 45) and (10, 10), we can use the formula:
y - y1 = m(x - x1)
where (x1, y1) represents one of the given points and m represents the slope.
First, let's find the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (3, 45) and (10, 10):
m = (10 - 45) / (10 - 3)
m = -35 / 7
m = -5
Now, substituting the values of (x1, y1) = (3, 45) and m = -5 into the point-slope equation:
y - 45 = -5(x - 3)
Simplifying and distributing -5:
y - 45 = -5x + 15
Finally, let's rearrange the equation into slope-intercept form:
y = -5x + 60
So, the point-slope equation for the given points is y - 45 = -5(x - 3).
To find the rate of change (slope), we can use the formula:
slope = (change in y) / (change in x)
In this scenario, the change in y is 60 feet - 10 feet = 50 feet, and the change in x is 10 days - 3 days = 7 days.
a. So, the rate of change (slope) can be calculated as:
slope = (50 feet) / (7 days) = 7.14 feet per day (rounded to two decimal places).
To find the y-intercept, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where (x1, y1) is any point on the line. We can use the point (10 days, 10 feet):
b. Plugging in the values, we get:
y - 10 = 7.14(x - 10)
Simplifying the equation:
y - 10 = 7.14x - 71.4
y = 7.14x - 61.4
c. The equation in slope-intercept form is: y = 7.14x - 61.4
To convert the equation into standard form (Ax + By = C), we move the variables to the same side of the equation:
7.14x - y = -61.4
d. The equation in standard form is: 7.14x - y = -61.4
To find the rate of change (slope), we can use the formula:
Rate of change (slope) = (change in y) / (change in x)
In this scenario, the change in y is the difference in the water level, and the change in x is the difference in the number of days.
Change in y = 10 ft - 45 ft = -35 ft
Change in x = 10 days - 3 days = 7 days
So, the rate of change (slope) can be calculated as follows:
Slope = (-35 ft) / (7 days) = -5 ft/day
The y-intercept is the value of y when x is zero. In this scenario, the river had a water level of 60 feet before the drought began. So, the y-intercept is 60 ft.
To write the equation for the scenario in slope-intercept form (y = mx + b), we can substitute the slope and y-intercept values we found:
y = -5x + 60
To write the equation in standard form (Ax + By = C), we need to rearrange the equation:
5x + y = 60
So, the answers are:
a. The rate of change (slope) for the scenario is -5 ft/day.
b. The y-intercept for the scenario is 60 ft.
c. The equation in slope-intercept form is y = -5x + 60.
d. The equation in standard form is 5x + y = 60.