Hello! Please help me, my school year ends this friday and I need to finish this in order to get course credit!!
Question: Come up with and describe two scenarios: one that models a direct variation situation and one that models an inverse variation situation. Do not state explicitly which scenario is which, but provide at least four data pairs for each situation. Your classmates will have to determine which of the scenarios is a direct variation and which is an inverse variation, and the value of k for each.
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Come up with and describe two scenarios: one that models a direct variation situation and one that models an inverse variation situation. Do not state explicitly which scenario is which, but provide at least four data pairs for each situation. Your classmates will have to determine which of the scenarios is a direct variation and which is an inverse variation, and the value of k for each.
can u help mee
Sure, here are two scenarios:
Scenario 1: The cost of a meal at a restaurant.
Data pairs:
| Number of people | Cost of meal |
| ---------------- | ------------ |
| 1 | $10 |
| 2 | $20 |
| 3 | $30 |
| 4 | $40 |
Scenario 2: The time it takes to complete a task with varying levels of effort.
Data pairs:
| Effort level | Time to complete task |
| ------------ | --------------------- |
| 2 hours | 10 minutes |
| 3 hours | 8 minutes |
| 5 hours | 6 minutes |
| 8 hours | 4 minutes |
It is up to your classmates to determine which scenario is a direct variation (where y=kx) and which scenario is an inverse variation (where y=k/x) and calculate the value of k for each.
Sure, I can help you with that! To come up with scenarios that model direct variation and inverse variation, we need to understand what each of these terms means in relation to mathematical relationships.
In direct variation, two variables are directly proportional to each other, meaning that as one variable increases, the other variable also increases at a constant rate. The general equation for direct variation is y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.
Now, let's create a scenario that models direct variation:
Scenario A:
Let's say you are filling up containers with water from a faucet. The time it takes to fill up the containers depends on the number of containers you have. Here are some data pairs:
Number of Containers (x): 1, 2, 3, 4
Time taken to fill up (y): 10 seconds, 20 seconds, 30 seconds, 40 seconds
In this scenario, the time taken to fill up the containers increases directly with the number of containers. You can see that for every additional container, the time taken also increases by 10 seconds. So, the equation for this scenario would be y = 10x, where k (the constant of variation) is 10.
Now let's move on to inverse variation:
Inverse variation occurs when two variables are inversely proportional to each other, meaning that as one variable increases, the other variable decreases, and vice versa. The general equation for inverse variation is y = k/x.
Scenario B:
Imagine you are driving a car at a constant speed on a stretch of road. The amount of time it takes to cover a certain distance depends on your speed. Here are some data pairs:
Speed (x): 60 mph, 50 mph, 40 mph, 30 mph
Time taken (y): 1 hour, 1.2 hours, 1.5 hours, 2 hours
In this scenario, the time taken to cover the distance decreases as the speed increases. As you can see, for every decrease in speed by 10 mph, the time taken increases by a certain factor. Using the equation y = k/x, we can determine the constant of variation. If we take the first data pair (60 mph, 1 hour), we have 1 = k/60. Solving for k, we get k = 60.
So, in scenario B, the equation would be y = 60/x, where k is 60.
By providing these two scenarios and the corresponding data pairs, you can now let your classmates determine which one is direct variation and which is inverse variation, as well as the value of k for each scenario.
I hope this explanation helps you understand how to create scenarios that model direct variation and inverse variation!