2.2+Keeping+Things+Balanced

Anant D
Big Idea: Many Real World Situations can be Modeled and Predicted using Mathematics Essential Question: What are some non-linear relationships?

Notes from Class:
Linear Relationships are relationships that have straight lines as their graphs, and have a constant rates of change in both the x and y axis. In this investigation, we discovered some data sets where there is a pattern in the tables and graphs. However, despite there being a pattern, there isn't a fixed rate of change in both the x and y values, and the graphs were usually curves. These relationships can be clasified as non-linear relationships.

Problem 2.2: Keeping Things Balanced
A.
 * Distance (cm) from fulcrum || Weight (Number of Weights) ||
 * 10 || 9 ||
 * 20 || 6 ||
 * 30 || 4 ||
 * 40 || 2.6 ||
 * 50 || 2.2 ||

B. Graph shown above. My graph tells us that as the distance from the fulcrum increases, the weights required to balance the meter stick decreases (hence the negative rate of change), as is also evident from my table in Part A. C. The graph and pattern of change in the data we found in this problem are similar to the ones we observed in Problem 2.1 as both graph models are curves (this shows both data sets did not have constant rates of change), and both data sets also had a decreasing rate of change, i.e as the x values increased, the y values decreased at an increasing rate. D. The data we collected in this Problem are different from the data we collected in Problem 1.1 and 1.2 as the relationship we observed in ths problem isn’t linear (in other words, it doesn’t have a straight graph nor a constant rate of change for both the x and y axis).


 * 2.2 Follow Up**

1. If we had done this experiment with different values to start off with, we would have gotten different results, though they would probably still exhibit the same general pattern. For example, if we used 2 weights at 20 cm, our balancing weights would've been higher, as 3 weights at 40 cm (the distance/weight) used in this experiment is equal to 1.5 weights at 20 cm, which is lighter than 2 weights at 20 cm. If we used 4 weights at 30 cm, we would've gotten significantly higher balancing weights, as 3 weights at 40 cm is equal to 2.25 weights at 30 cm, which is once again lower than 4 weights at 30 cm. If we used 2 weights at 50 cm, we would've gotten much lower balancing weights, as 3 weights at 40 cm is equal to 3.75 weights at 50 cm, which is higher than 2 weights at 50 cm. However, as I mentioned earlier, all the results we would've obtained from using the different weights shown above would've still exhibited a similar pattern to the one we observed from our results. 2. Equations B, E, and F satisfy the data pairs (30,4), (24,5), and (20,6)

Equation B: W = 120/D - 4 = 120/30 becomes 4 = 4 - 5 = 120/24 becomes 5 = 5 - 6 = 120/20 becomes 6 = 6

Equation : D = 120/W - 30 = 120/4 becomes 30 = 30 - 24 = 120/5 becomes 24 = 24 - 20 = 120/6 becomes 20 = 20

Equation E: WD = 120 - 4 X 30 = 120, which is an accurate statement - 5 X 24 = 120, also an accurate statement - 6 X 20 = 120, which is also correct

Equation F: DW We know that this equation also satisfies the three data pairs as this eaquation is essentially the same as Equation E (we can use the commutative property of multiplication to flip WD)