I am totally stuck on this question, can some one help?
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.
Direct:
y = kx
As x increases, y increases
distance = rate(time)
y/x = k
miles/time = rate
100, 5
200, 10
400, 20
800, 40
Inverse xy = k
As x increases, y decreases
As pressure increases on a gas, the volume decreases.
100 2
200 1
400 1/2
800 1/4
Thanks John1!
which one is right for each.?
Sure, I can help you with that. To come up with scenarios that model direct variation and inverse variation, we first need to understand what they mean.
In direct variation, when one variable increases, the other variable also increases. The relationship between the two variables can be expressed as y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation. In direct variation, as x increases, y increases at a constant rate determined by k.
In inverse variation, when one variable increases, the other variable decreases. The relationship between the two variables can be expressed as y = k/x, where y is the dependent variable, x is the independent variable, and k is the constant of variation. In inverse variation, as x increases, y decreases at a rate determined by k.
Now, let's create two scenarios:
Scenario 1: Direct Variation
Let's consider the number of hours studied (x) and the score obtained (y) on a test.
Data pairs for Scenario 1 (Direct Variation):
(1, 80)
(2, 160)
(3, 240)
(4, 320)
In this scenario, as the number of hours studied (x) increases, the score obtained (y) also increases. The constant of variation, k, can be determined by taking any data pair (x, y) and dividing the y-value by the x-value. For example, using the first data pair (1, 80), we get 80/1 = 80. Therefore, k = 80.
Scenario 2: Inverse Variation
Let's consider the number of workers (x) on a construction project and the time taken (y) to complete the project.
Data pairs for Scenario 2 (Inverse Variation):
(2, 20)
(4, 10)
(6, 6.67)
(8, 5)
In this scenario, as the number of workers (x) increases, the time taken to complete the project (y) decreases. The constant of variation, k, can be determined similarly by taking any data pair (x, y) and multiplying the y-value by the x-value. For example, using the first data pair (2, 20), we get 2 * 20 = 40. Therefore, k = 40.
Remember, in direct variation, the equation is y = kx, while in inverse variation, the equation is y = k/x. In direct variation, as x increases, y increases, while in inverse variation, as x increases, y decreases.
I hope this helps you understand the concept of direct variation and inverse variation and how to come up with scenarios for each.