A new toilet model has two different flush settings in order to conserve water. One setting uses 1.6 gallons per flush and the other setting uses .8 gallons per flush. The total amount of water w (in gallons) used in the first setting is given by the equation w=1.6f where f is the number of times the toilet is flushed. The total amount of water used in the second setting is given by the equation w=.8f.
A) Graph both equations in the same coordinate plane. What do the slopes and the w intercepts mean in this sisuation?
B) How much more water is used by the first setting if the toilet is flushed ten times.
these are both straight lines through the origin (intercepts both = 0), one with slope 1.6 g/flush and the second with slope 0.8 g/flush
w = 1.6*10 = 16 gal
w = 0.8*10 = 8 gal
16 - 8 = 8 gal more
A) To graph the equations w = 1.6f and w = 0.8f in the same coordinate plane, we'll plot points for different values of f.
For w = 1.6f:
If f = 0, then w = 0 (0,0)
If f = 1, then w = 1.6 (1,1.6)
If f = 2, then w = 3.2 (2,3.2)
And so on...
For w = 0.8f:
If f = 0, then w = 0 (0,0)
If f = 1, then w = 0.8 (1,0.8)
If f = 2, then w = 1.6 (2,1.6)
And so on...
The graph having w on the y-axis and f on the x-axis would look like this:
|
4 | *
| * .
3 | . . . . . . . . .
| .
2 | .
| * *
1 | .
| *
0 |_____________________
0 1 2 3 4
f
The blue line on the graph represents the equation w = 1.6f, and the red line represents w = 0.8f.
The slopes of the lines represent the rate of change of water used per flush. In this case, the slope of the blue line (1.6) means that for every additional flush, the water usage increases by 1.6 gallons. Similarly, the slope of the red line (0.8) means that for every additional flush, the water usage increases by 0.8 gallons.
The w-intercept (where the lines cross the y-axis) represents the initial amount of water used when f = 0. In this case, both lines intersect at the origin (0,0), meaning that when no flushes have occurred, no water is used.
B) To find out how much more water is used by the first setting compared to the second after ten flushes, we can simply substitute f = 10 into both equations and calculate the difference:
For the first setting, w = 1.6f:
w = 1.6 * 10 = 16 gallons
For the second setting, w = 0.8f:
w = 0.8 * 10 = 8 gallons
The first setting uses a total of 16 gallons, while the second setting uses a total of 8 gallons. Therefore, the first setting uses 16 - 8 = 8 more gallons of water when the toilet is flushed ten times.
A) To graph both equations on the same coordinate plane, we can plot the points that satisfy each equation and draw a line through these points.
For the first equation, w = 1.6f, we can choose a few values for f and calculate the corresponding values for w. Let's use f = 0, 1, 2, and 3:
For f = 0, w = 1.6 * 0 = 0 gallons.
For f = 1, w = 1.6 * 1 = 1.6 gallons.
For f = 2, w = 1.6 * 2 = 3.2 gallons.
For f = 3, w = 1.6 * 3 = 4.8 gallons.
Plotting these points on the graph, we get a linear line with a positive slope. The slope of the line is 1.6, which means that for every increase of 1 in f, there is an increase of 1.6 in w. The w-intercept is at (0, 0), indicating that if the toilet is not flushed (f = 0), no water is used (w = 0).
For the second equation, w = 0.8f, we can follow the same process:
For f = 0, w = 0.8 * 0 = 0 gallons.
For f = 1, w = 0.8 * 1 = 0.8 gallons.
For f = 2, w = 0.8 * 2 = 1.6 gallons.
For f = 3, w = 0.8 * 3 = 2.4 gallons.
Plotting these points on the graph, we get another linear line with the same positive slope. The slope of the line is 0.8, indicating that for every increase of 1 in f, there is an increase of 0.8 in w. The w-intercept is also at (0, 0), indicating that if the toilet is not flushed (f = 0), no water is used (w = 0).
B) To find out how much more water is used by the first setting if the toilet is flushed ten times, we can calculate the amount of water used for each setting and find the difference.
For the first setting, using w = 1.6f, we substitute f = 10:
w = 1.6 * 10 = 16 gallons.
For the second setting, using w = 0.8f, we substitute f = 10:
w = 0.8 * 10 = 8 gallons.
The difference in water usage between the two settings is:
16 gallons - 8 gallons = 8 gallons.
Therefore, the first setting uses 8 more gallons of water when the toilet is flushed ten times compared to the second setting.