13Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rainThe river was 60 feet before the drought began (points)
a What is the rate of change slopefor this scenario?
bWhat is the y-intercept for this scenario?
cWhite an equation for the scenario in slope-intercept form
d White an equation in standard form
a) The rate of change (slope) for this scenario can be calculated by finding the change in the y-coordinate (water level) divided by the change in the x-coordinate (days).
Change in water level = 45 ft - 10 ft = 35 ft
Change in days = 3 days - 10 days = -7 days (negative because the days are decreasing)
Rate of change (slope) = Change in water level / Change in days
= 35 ft / (-7 days)
= -5 ft/day
b) The y-intercept for this scenario can be determined by looking at the initial water level, which was 60 feet.
The y-intercept is the point where the line crosses the y-axis, which corresponds to the value of the dependent variable (water level) when the independent variable (days) is zero. In this case, when there are zero days of no rain, the water level is 60 feet.
Therefore, the y-intercept is 60.
c) The equation for this scenario in slope-intercept form is y = mx + b, where m represents the slope and b represents the y-intercept.
Plugging in the values we found:
y = -5x + 60
d) To convert the equation to 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 of the equation.
Starting with the equation in slope-intercept form:
y = -5x + 60
Rearranging to move the x term to the left side:
5x + y = 60
Switching the sides to have the constant term on the right side:
5x + y - 60 = 0
So, the equation in standard form is:
5x + y - 60 = 0
To find the rate of change (slope), we need to use the formula:
Rate of change (slope) = (change in y) / (change in x)
a) The change in y is 45 feet (the water level after 3 days) - 10 feet (the water level after 10 days) = 35 feet.
The change in x is 3 days - 10 days = -7 days.
Rate of change (slope) = 35 feet / -7 days = -5 feet/day
b) The y-intercept is the value of y when x is zero. In this scenario, it represents the initial water level before the drought began. The initial water level is 60 feet.
c) To write the equation in slope-intercept form (y = mx + b), we can use the slope and y-intercept values we found:
y = -5x + 60
d) To write the equation in standard form (Ax + By = C), where A, B, and C are integers:
Rearranging the slope-intercept equation:
5x + y = 60
Multiplying both sides by -1 to make the leading coefficient positive:
-5x - y = -60
Alternatively, we can multiply the equation by -1 from the beginning:
-5x - y = 60
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 represents the change in the water level of the river, and the change in x represents the number of days without rain.
a) To find the rate of change (slope), we can calculate:
Slope = (10 feet - 45 feet) / (10 days - 3 days)
= -35 feet / 7 days
= -5 feet per day
Therefore, the rate of change (slope) for this scenario is -5 feet per day.
b) The y-intercept represents the initial value of the y-variable when x is equal to zero. In this case, the y-intercept is the water level of the river before the drought began, which is 60 feet.
c) To write an equation for this scenario in slope-intercept form (y = mx + b), we can substitute the values we found:
y = -5x + 60
Therefore, the equation representing the water level of the river in slope-intercept form is y = -5x + 60.
d) To write the equation in standard form (Ax + By = C), we can rearrange the equation:
5x + y = 60
Therefore, the equation representing the water level of the river in standard form is 5x + y = 60.