Rate of Change

1. Rate of Change

2. But first… a quick Math review: When solving equations: Step One: Write down what you know/What you are given in the question Step Two: Pick and write down the correct formula (one where you know everything but one variable) Step Three: Using the same format of the equation, fill in the proper numbers in the right places Step Four: Solve, and include proper units

3. Time Intervals • ∆t = tfinal – tinitial • ∆ = delta or “change in” • “final” means the time you ended • “initial” means the time you started • You can find the tfinal ortinitial using these formulas: • tfinal = ∆t + tinitial • tinitial = tfinal – ∆t

4. Example One: If you left home at 7:33 a.m. and arrived to school at 8:05 a.m., how long did it take you? ∆t = tfinal – tinitial tfinal = 8:05 a.m. ∆t = 8:05 – 7:33 tinitial = 7:33 a.m ∆t = 32 min. .

5. Example Two: If a runner on a track passed you at 39 seconds, and it takes him 54 seconds to run the length of the track, at what time will he pass you by a second time? tfinal = ∆t + tinitial ∆t = 54 sec. Tfinal = 54 sec. + 39 sec.Tinitial = 39 sec. Tfinal = 93 sec. or 1 min. 33 sec.

6. Distance Intervals • ∆d = dfinal – dinitial • “final” means the distance at which you ended • “initial” means the distance at what you started • You can find the dfinal ordinitial using these formulas: • dfinal = ∆d + dinitial • dinitial = dfinal – ∆d

7. Example One: If you left home and walked for 30 minutes and traveled a distance of 2500 metres, how far did you walk? ∆d = dfinal – dinitial dfinal = 2500 m. ∆d = 2500 – 0 dinitial = 0 m. ∆d = 2500 m .

8. Example Two: What was your initial position if you ran for 850 m and ended at a position of 1387 m? dinitial = dfinal – ∆d dfinal = 1387 m dinitial = 1387 – 850∆d = 850 m dinitial = 537 m

9. Rate of Change of a Linear Relationship run rise Rate of Change= The rate of change of a linear relationship is the steepnessof the line.

10. Rates of change are seen everywhere. Engineers refer to the steepness of the roof of a house as the pitch Engineers refer to the rate of change of a road as the grade

11. Engineers often represent the rate of change as a percentage.

12. 100 8 Rate of change = A grade of 8% would mean for every rise of 8 unitsthere is arun of 100 units. = 8%

13. The steepness of wheelchair ramps is of great importance for safety. 1 12 Rate of change of wheelchair ramp = If the rise is 1.5 m, what is the run? X 1.5 Answer: 18 m because

14. Determine the rate of change (pitch) of the roof. 3 m 5 m

15. 3 2 3 3 Determine the rate of change of each staircase.

16. Determine the rate of change. Which points will you use to determine rise and run? Earnings 4 20 = $5/hr What does this rate of change represent? Number of Hours Worked The hourly wage

17. Rate of Change Calculations Rate of change (r) = ∆ x ÷ ∆t ∆ x = xfinal – xinitial ∆t = tfinal - tinitial

18. Example One: At 6:00 a.m. the temperature is 2 degrees. At 10:00 a.m. the temperature is 14 degrees. What is the average rate of change in the temperature? r = ∆ x ÷ ∆t r = (14 – 2) ÷ (10:00 – 6:00) r = 12 ÷ 4 r = 3 degrees/hour

19. Example Two: Ron weighed 64 kg in 2008. He weighed 52 kg in 2012. What was his rate of change in his weight? r = ∆ x ÷ ∆t r = (64 – 52) ÷ (2012 – 2008) r = 12 ÷ 4 r = 3 kg/year