Advance Proportions Unit 1 - Science

Advanced Proportions Sometimes the variables in a proportion have… exponents

Advanced Proportions If Y is directly proportional to X then… …If X doubles, Y will _______________. 2 Quadruple

Equation: Y= 2 k X

How do I solve for K? Graph Y vs. 2 X & the slope = k

Natural Graph vs. Linearized Graph LINERIZED GRAPH: Y Y X x 2

Example: You are curious if there is a relationship between the stretch distance of a tennis-ball launcher and how high the ball goes. You play with a launcher all afternoon and record the following data…

Here is the data you collect: Stretch (m) Height (m) .1 1.5 .2 6 .3 15 .4 26 .5 42

Graph the information on the first graph grid in your notes.

Your graph should look similar to this: How the stretch distance affects the height of a tennis-ball Height (m) Stretch (m)

Now we need to linearize the graph: To do this we will graph X2 and Y. First we need to change our data table.

Our new data table will look like this: X2 Y (.1)2 = .01 1.5 (.2)2 = .04 6 (.3)2 = .09 15 (.4)2 = .16 26 (.5)2 = .25 42

Graph the information on the second graph grid in your notes.

Your graph should look similar to this: X2

Next Steps: Write the EQUATION FORM: Y= kx2 Circle your cross points. Solve for K. K = (50-25)/(.30-.15) = 25/.15 = 166.67

So our real world equation is: H = 166.67 s 2 Where H is the height & s is the stretch distance.

Summing it up! Based on the shape of your graph, the height of the ball is directly proportional to stretch2 . This means if you stretch THREE times longer you will launch 9 times higher! Use your real world equation to predict how high the tennis ball will go if you stretch the launcher 0.75 meters. Answer: 93.75 meters high

Now try it on your own! Go back to your car graph from the basic graphs lab and determine the real world equation.

Advanced Proportions If Y is inversely proportional to X then… …If X doubles, Y will _______________. 2 1/4

Equation: Y= k __ X2

How do I solve for K? Graph Y vs. 1 __ X2 & the slope = k

Natural Graph vs. Linearized Graph Y Y X 1/X2

Example: You are curious if there is a relationship between the intensity of sunlight and the distance a planet is from the sun. You send out space probes and collect the following data…

Here is the data you collect: Distance (AU) Sunlight (W/m2 ) 0.4 6800 1 986 1.5 438 5.2 36 40 0.6

Graph the information on the first graph grid in your notes.

Your graph should look similar to this: Sunlight vs. Planet Distance Sunlight (W/m2 ) Distance (AU)

Now we need to linearize the graph: To do this we will graph 1/x2 and Y. First we need to change our data table.

Our new data table will look like this: 1/x2 Y 1/(.4)2 = 6.25 6800 1/(1)2 = 1 986 1/(1.5)2 = 0.44 438 1/(5.2)2 = 0.04 36 1/(40)2 = 0.000625 0.6

Graph the information on the second graph grid in your notes.

Your graph should look similar to this: 1/x2

Next Steps: Write the EQUATION FORM: Y= k/x2 Circle your cross points. Solve for K. K = (4200-2100)/(4-2) = 2100/2 = 1050

So our real world equation is: 1050 ____ d2 Where S is the sunlight & d is the distance.

Summing it up! Based on the shape of your graph, the intensity of sunlight is inversely proportional to distance2 . This means if a planet is FOUR times further from the sun, the light will be 1/16 times as bright! Use your real world equation to predict how much sunlight Venus gets if it is 0.72 AU from the sun. Answer: 2025.46 W/m2

Now answer the questions to the real world application at the bottom of the page.

Now let’s do a real world application Now let’s do a real world application. Follow the instructions for the Slinky Lab found on the previous page.