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)
a. What is the rate of change (slope) for this scenario?
b. What is the y-intercept for this scenario?
c. Write an equation for the scenario in slope-intercept form.
d. Write this equation in standard form.
(0,34)
y = 4x + 34
(0, 4)
y = 4x - 22
- 4x + y = 34
y = 4x + 22
(0.0)
(0,22)
4x + y = 34
7
22
28
n - 4x + u = 22
v = 28x + 34
0
4
a. The rate of change (slope) for this scenario is 4.
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.
To find the answers, we can use the given information about the water level at different times.
a. The rate of change (slope) can be found by calculating the change in y divided by the change in x. In this case, the change in y is 62 - 34 = 28 and the change in x is 10 - 3 = 7. Therefore, the rate of change is 28/7 = 4.
b. The y-intercept is the value of y when x is equal to 0. From the given information, we know that when x is 0, y is 22. Therefore, the y-intercept is 22.
c. To write the equation in slope-intercept form (y = mx + b), we can substitute the slope and the y-intercept into the equation. This gives us y = 4x + 22.
d. To write the equation in standard form (Ax + By = C), we move all the terms to one side of the equation. This gives us 4x - y = -22.
So, the answers are:
a. The rate of change (slope) is 4.
b. The y-intercept is 22.
c. The equation in slope-intercept form is y = 4x + 22.
d. The equation in standard form is 4x - y = -22.
a. To find the rate of change (slope) for this scenario, we need to determine the change in the water level divided by the change in the number of days.
The change in the water level is 62 feet - 34 feet = 28 feet.
The change in the number of days is 10 days - 3 days = 7 days.
So, the slope is 28 feet / 7 days = 4 feet per day.
b. The y-intercept represents the initial water level before the rain began. In this scenario, the river was 22 feet before the rain started. Therefore, the y-intercept is 22 feet.
c. To write an equation for the scenario in slope-intercept form (y = mx + b), we can use the slope and y-intercept we found in parts (a) and (b). The equation will be:
y = 4x + 22
d. To write this equation in standard form (Ax + By = C), we can rearrange the equation:
y - 4x = 22
Multiply both sides by -1 to make the coefficient of x positive:
-1(y - 4x) = -1(22)
Distribute the negative sign:
- y + 4x = -22
Finally, move all the terms to one side:
-4x + y = -22
So, the equation in standard form is -4x + y = -22.