Follow the instructions below to design a fair race for the new video game Animal Tracks.
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.
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.
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.
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.
Your equations, graph, and explanation for your race design will be submitted as your portfolio assessment.
To design a fair race for the video game Animal Tracks, you need to choose two animals with different speeds and create a race where they have an equal chance of winning if they race at their top speed. Here's how you can do it:
Step 1: Choose two animals with different speeds.
You can either choose from the chart provided or do some research to choose your own animals. Let's say you choose a Cheetah and a Tortoise. The Cheetah is much faster than the Tortoise.
Step 2: Design the race.
To make the race fair, you can give the slower animal (Tortoise) a shorter distance to race. This compensates for the speed difference between the two animals.
Step 3: Create a system of two linear equations.
Let's define the following variables:
- Let d1 represent the distance the Cheetah will run in the race.
- Let d2 represent the distance the Tortoise will run in the race.
- Let s1 represent the speed of the Cheetah.
- Let s2 represent the speed of the Tortoise.
Since the race is fair when both animals finish the race in the same amount of time, you can use the equation:
Time taken by Cheetah = Time taken by Tortoise
The time taken can be calculated using the formula: Time = Distance / Speed.
So, we get the following equations:
d1 / s1 = d2 / s2
Step 4: Graph the system to prove the race is fair.
To graph the system of equations, you can convert the equations to slope-intercept form (y = mx + b).
Equation 1: d1 / s1 = d2 / s2
Rearranging, we get:
d1 / s1 - d2 / s2 = 0
Multiplying both sides by s1 * s2, we get:
d1 * s2 - d2 * s1 = 0
This equation represents the race conditions.
Graphing this equation will give us a line. The slope of this line represents the relative speeds of the two animals. The fact that the line intersects the x-axis (d1) means that for equal distances covered by the Cheetah and Tortoise, their times will be equal - indicating a fair race.
Equation 2: Let's assume the maximum distance of the race is D. Since the slower animal (Tortoise) has a shorter distance to race, let's say it covers only half the distance, so d2 = D/2.
Now we can substitute this value into Equation 1:
d1 * s2 - (D/2) * s1 = 0
This equation will give us another line when graphed.
By graphing these two equations, you can visually see where they intersect. The intersection point on the graph represents the fair race condition, where both animals have an equal chance of winning if they race at their top speeds.
To summarize, to design a fair race for the video game Animal Tracks:
1. Choose two animals with different speeds.
2. Give the slower animal a shorter distance to race.
3. Create a system of two linear equations to represent the race conditions.
4. Graph the equations to prove that the race is fair.