Directions: Follow the instructions below to design a fair race for the new video game Animal Tracks.
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. (I chose elephant which is 25 mph and an ostrich which is 40 mph.)
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.
Your equations, graph, and explanation for your race design will be submitted as
your portfolio assessment.
Can someone give me an example of the exact equation and graphs I would use? I already have written down how one animal would win (my explanation) the race but I don't know how to write it as an equation and graph :(
Did you ever get the answer to this i need a bunch of help
This is actually Pre-Algebra, not just math if that makes a difference.
To design a fair race between an elephant (25 mph) and an ostrich (40 mph), we need to ensure that both animals have an equal chance of winning the race. The steps below outline how to create equations and graph to demonstrate this.
Step 1: Choose two animals with different speeds:
- Elephant: Speed = 25 mph.
- Ostrich: Speed = 40 mph.
Step 2: Design a fair race:
To create a fair race, we will give the slower animal (elephant) a shorter distance to travel. Let's assume the race is represented by a straight line, and we will denote the distance that each animal will travel as D. Since the animals will be racing at their top speed for the entire distance, the time taken by each animal to complete the race will be the same.
Step 3: Write a system of two linear equations:
Let's assume the elephant's race distance is denoted by DE, and the ostrich's race distance is denoted by DO. We can write the following system of equations to represent the fair race:
Elephant: Distance = Speed × Time
DE = 25T
Ostrich: Distance = Speed × Time
DO = 40T
Here, T represents the time taken by both animals to complete the race, which will be equal.
Step 4: Graph the system:
To graph this system of equations, we can plot the distances traveled by each animal on the y-axis and the time taken on the x-axis.
Let's assume the maximum time taken for the race is 1 hour. We can choose different values for T and calculate the corresponding distances for both animals.
For example, assume T = 0.5 hour:
Elephant: DE = 25T = 25 × 0.5 = 12.5 miles
Ostrich: DO = 40T = 40 × 0.5 = 20 miles
Plot these points on the graph, with 12.5 on the y-axis for the elephant and 20 on the y-axis for the ostrich. Connect the two points with a line.
Repeat this process for different values of T, and you will notice that the lines representing the distances traveled by the elephant and ostrich are parallel. This indicates that for any value of T (time), the distance traveled by the elephant will always be less than the distance traveled by the ostrich.
Explanation:
Since the lines representing the distances are parallel, it means that there is no point where the elephant will cross or overtake the ostrich during the race. This ensures that both animals have an equal chance of winning, as the timing of their finish will solely depend on their speeds and the initial distance given to each animal.
The fairness of the race is proven by the graph, which demonstrates that regardless of different values of T (time), the elephant's distance will always be less than the ostrich's distance, indicating a shorter distance for the elephant but compensating for it with its slower speed.
To design a fair race for the video game Animal Tracks, you can use the given speeds of the animals and create a system of linear equations to model the race. Let's use your chosen animals, the elephant (with a speed of 25 mph) and the ostrich (with a speed of 40 mph).
Step 1: Create the equations to represent the distances each animal can travel in the same amount of time.
Let's assume the race distance for the ostrich is "x" (in any unit) and the race distance for the elephant is "y" (also in the same unit). Since the race is fair if the animals finish at the same time, their travel times should be equal.
The time it takes for the ostrich to complete the race is given by the formula:
Time = Distance / Speed
=> Time = x / 40
Similarly, the time it takes for the elephant to complete the race is:
Time = Distance / Speed
=> Time = y / 25
Since the travel times are equal, we can set up an equation:
x / 40 = y / 25
This equation represents the condition that both animals finish the race in the same amount of time.
Step 2: Graph the system of equations.
To graph the system of equations, we can rearrange the equation from Step 1 into slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Let's rearrange the equation:
x / 40 = y / 25
Multiply both sides by 40 to eliminate the fraction:
x = (40 / 25) * y
x = (8 / 5) * y
Now we can graph the equation y = (5/8)x:
y-axis intercept: (0, 0)
x-axis intercept: (5, 8)
Plot these two points and draw a straight line passing through them.
Step 3: Explain how the graph proves the race is fair.
On the graph, every point on the line represents a possible pair of distances (x, y) that would ensure a fair race. The line shows all the combinations of distances for the ostrich and elephant in which they will finish the race in the same amount of time.
If the coordinates of any point on this line are (x, y), it means that if the race distance for the ostrich is x, then the race distance for the elephant would be y. And if both animals run at their designated speeds, they will finish the race at the same time.
By designing the race based on this equation and graph, you have ensured a fair competition for the animals in the video game Animal Tracks.