1. Choose two animals with different speeds. You can choose from the chart that starts at the bottom of this page or do research to choose your own.
2. Design a fair race in which the two animals 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.
I am not going to put the graph since I picked my animals,
Ostrich- 40 mph
Lion- 50 mph
A fair race between an ostrich and a lion would be a race of 200 meters. The ostrich would have to travel 200 meters and the lion would have to travel 160 meters. This would give the ostrich and the lion an equal chance of winning the race if they both ran at their top speed.
The system of linear equations to model the fair race would be:
Ostrich: x = 40t
Lion: y = 50t
Where x is the distance the ostrich travels and y is the distance the lion travels, and t is the time it takes for each animal to complete the race.
The graph of the system of linear equations would be two lines that intersect at the point (200, 160). This graph proves that the race is fair because it shows that the two animals will reach the finish line at the same time if they both run at their top speed.
1. I have chosen the Ostrich and the Lion as my two animals with different speeds. The Ostrich has a speed of 40 mph, while the Lion has a speed of 50 mph.
2. To design a fair race, I will give the slower animal, which is the Ostrich, a shorter distance to race. This ensures that both animals have an equal chance of winning if they race at their top speed.
3. Let's define some variables:
- Let dO be the distance the Ostrich travels.
- Let dL be the distance the Lion travels.
- Let vO be the speed of the Ostrich, which is 40 mph.
- Let vL be the speed of the Lion, which is 50 mph.
The time it takes for both animals to complete the race is the same, as it's a fair race. The time can be calculated using the formula: time = distance / speed.
For the Ostrich: time = dO / vO
For the Lion: time = dL / vL
Since the time for both animals is the same, we can set up the following equation:
dO / vO = dL / vL
4. To graph the system, let's create a coordinate plane. The x-axis will represent the distance the Ostrich travels (dO), and the y-axis will represent the distance the Lion travels (dL).
Now, we can solve the equation for dL:
dL = (vL / vO) * dO
Let's substitute the values of the speeds: vL = 50 mph and vO = 40 mph. The equation becomes:
dL = (50 / 40) * dO
dL = (5/4) * dO
Now, we can plot some points on the graph to see how the two animals' distances relate to each other. For example, if the Ostrich travels 80 miles (dO = 80), the Lion would travel:
dL = (5/4) * 80
dL = 100
Plotting the point (80, 100) on the graph, we can see that it lies on a straight line. By plotting more points, we can see that all the points lie on a straight line.
The graph demonstrates that regardless of the distance traveled by the Ostrich, the distance traveled by the Lion is always 1.25 times that of the Ostrich. This ensures that both animals have an equal chance of winning the race.
By designing the race in this way, we have created a fair competition where the slower animal has a shorter distance to cover, giving both animals an equal opportunity to win if they race at their top speeds.
To design a fair race between an ostrich and a lion, we need to consider their different speeds. The ostrich has a speed of 40 mph, while the lion has a speed of 50 mph.
To make the race fair, we can give the slower animal, which is the ostrich, a shorter distance to race. Let's assume that the lion runs at its top speed for a certain distance, while the ostrich runs at its top speed for a shorter distance.
Let:
- d1 be the distance the lion runs (in miles)
- d2 be the distance the ostrich runs (in miles)
- t1 be the time it takes for the lion to run the race (in hours)
- t2 be the time it takes for the ostrich to run the race (in hours)
- s1 be the lion's speed (50 mph)
- s2 be the ostrich's speed (40 mph)
We know that speed is equal to distance over time. Therefore, we can set up the following equations:
For the lion:
s1 = d1 / t1
For the ostrich:
s2 = d2 / t2
Since we want both animals to finish the race in the same amount of time, we can set t1 = t2.
Now, let's solve this system of equations to find the values of d1 and d2.
From the equation for the lion:
t1 = d1 / s1
d1 = t1 * s1
From the equation for the ostrich:
t2 = d2 / s2
d2 = t2 * s2
Since t1 = t2, we can substitute t1 with t2 in the equation for the lion:
d1 = t2 * s1
Now, we need to compare the distances traveled by the lion and the ostrich.
If the lion takes the same amount of time as the ostrich, we can write an equation using the distances traveled:
d1 = d2
Substituting the expressions for d1 and d2 from above, we get:
t2 * s1 = t2 * s2
Since s1 > s2 (50 mph > 40 mph), the time t2 must be smaller than the time t1 for the lion to cover the same distance as the ostrich. This condition ensures that the race is fair.
Graphically, if we plot the distance (vertical axis) versus time (horizontal axis) for both animals, we will see that the lion's line is steeper than the ostrich's line. The crossing point of these lines represents the fair race condition, where both animals would finish the race at the same time. The graph confirms that the race is fair since the crossing point represents the equal chance of winning for both animals.
By using the values given for the speeds of the ostrich (40 mph) and the lion (50 mph), you can plot the graph to illustrate the fair race.