Write a system of two linear equations showing the distance a lion and cheetah at a constant top speed of 50 miles per hour for the lion and 70 for the cheetah. The distance they must run is 140 miles for the cheetah and 100 miles for the lion.
Let's use x to represent the time in hours that both the lion and cheetah have been running, and let's use y to represent the distance each of them has covered.
For the lion:
y = 100 + 50x
For the cheetah:
y = 140 + 70x
Both equations represent a linear relationship between distance and time, with the slope being their respective top speeds (50 for the lion, 70 for the cheetah) and the y-intercept being the distance they had covered before they started running (100 miles for the lion, 140 miles for the cheetah).
2. Design a fair race in which a cheetah and a liion have an equal chance of winning
if they race at their top speed. Here are a few tips for your design:
a. The race is fair if the two animals could finish the race in the same
amount of time.
b. You can give the slower animal a shorter distance to race.
c. Since this is a video game, the race does not need to be realistic—it
can be any length, and the animals can run at a constant speed.
3. Write a system of two linear equations showing the distance each animal can
travel to model the fair race. Be sure to define all variables.
4. Graph the system to prove that the two animals have an equal chance of
winning the race. Explain how the graph proves the race is fair.
3. System of linear equations:
Let's use x to represent the time in hours that both animals have been running, and y to represent the distance each animal has covered.
Let dL be the distance covered by the lion and dC be the distance covered by the cheetah.
Then we have:
dL = 50x
dC = 70x - 20
The first equation represents the distance covered by the lion, which is equal to its speed (50 mph) times the time it has been running (x).
The second equation represents the distance covered by the cheetah, which is equal to its speed (70 mph) times the time it has been running (x), minus a constant displacement of 20 miles, to offset the advantage of its higher top speed.
4. Graph:
Let's start by graphing the equation for the lion:
y = 50x
This is a straight line passing through the origin (0,0) with a slope of 50/1 = 50.
Now let's graph the equation for the cheetah:
y = 70x - 20
This is also a straight line with a slope of 70/1 = 70, but with a y-intercept of -20.
The graph shows that both animals will cover the same distance at the same time if they cover a distance of 400 miles.
To see why, notice that the point at which the two lines intersect is (0.4, 20). This means that if both animals run for 0.4 hours, the lion will have covered 20 miles and the cheetah will have covered 0 miles, giving the lion a head start of 20 miles. However, since the cheetah runs faster, it will catch up to the lion exactly when both animals have covered a distance of 400 miles, as shown by the intersection point.
Therefore, if the race is set at a distance of 400 miles, the lion and the cheetah will have an equal chance of winning, assuming that they run at their top speed of 50 mph and 70 mph respectively, and that they both start running at the same time.
To write a system of linear equations showing the distance traveled by the lion and cheetah at a constant top speed, we can use the formula:
Distance = Speed × Time
Let's assume the time for which both the lion and cheetah travel is the same, represented by 't'.
For the lion:
Distance (D1) = Speed of the lion (S1) × Time (t)
D1 = 50t
For the cheetah:
Distance (D2) = Speed of the cheetah (S2) × Time (t)
D2 = 70t
Given that the distance the cheetah must run is 140 miles and the distance the lion must run is 100 miles, we can set up the following equations:
D1 = 100
D2 = 140
Substituting the formulas for distance derived above, we have:
50t = 100
70t = 140
Thus, the system of equations representing the given scenario is:
50t = 100
70t = 140