40. What does it mean if two quantities vary directly? What it means is that when two variables are related in this kind of way the ratio of their values always remain the same which the two variables are said to be in direct variation.
41. In your own words, explain how to solve a variation problem. When attempting to solve the best way to go about it would be to follow the 4 step process. The first step would be to write down the selected equation. All direct variation problems are solved by using the equation y=kx. The second step would be to use the information given in the problem to find the value of K which is called the constant of variation or the constant of proportionality. The next step in this process would be to rewrite the equation from step one substituting in the value of k found in step two. The final step would be to make use of the equation found in step three and the remaining information given in the problem to answer the question asked also when solving word problems it helps to include units in your final answer as well.
42. What does it mean if two quantities vary inversely? With two quantities with inverse variation as one quantity increases the other quantity decreases.
43. Explain what is meant by combined variation. Give an example with your explanation. Combined variation involves a combination of direct or joint variation and indirect variation. For example if y varies directly with x and inversely with z, and y = 25 when x = 10 and z = 2, find y when x = 18 and z = 9. | | First, write the general form for combined variation: Plug in the given values and solve for k:Cross multiply and solve for k:50 = 10kk = 5 Now plug our k value into the general equation: Now find y when x = 18 and z = 9 y = 10 | |
44. Explain what is meant by joint variation. Give an example with your explanation. A joint variation is where two variables are related directly. My given example of a joint variation -z is jointly proportional to x and y and z = 6, when x = 3 and y = 4, find z when x = 7 and y = 4.
Find k:
6 = 3(4)k
Then, find z when x = 7 and y = 2.
45. Joint variation
46. Inverse Variation
47. We have seen that the daily number of phone calls between two cities varies jointly as their populations and inversely as the square of the distance between them. This model, used by telecommunication companies to estimate the line capacities needed among various cities, is called the gravity model.
Compare the model to Newton’s formula for gravitation on page 400 and describe why the name gravity model is appropriate. These models relate in very similar in ways but the Newton’s gravity formula model is more constant since the moon passes over the ocean every night of every day. When overlooking telecommunication companies this has the same process but not as constant but the gravity is compared to the actual population and the distance between the various cities.
