A rate of change is a ratio of the amount of change in the dependent variable to the amount of change in the independent variable.
Example 1 :
The graph shows the distance Nathan bicycled over time. What is Nathan’s rate of change ?
Solution :
Step 1 :
Identify the independent and dependent variables.
Independent : Time
Dependent : Distance
Step 2 :
Find the rates of change.
0 hour to 1 hour :
Change in distance/Change in time = (15 - 0)/(1 - 0)
Change in distance/Change in time = 15/1
Change in distance / Change in time = 15
1 hours to 4 hours :
Change in distance/Change in time = (60 - 15)/(4 - 1)
Change in distance/Change in time = 45/3
Change in distance/Change in time = 15
2 hours to 4 hours :
Change in distance/Change in time = (60 - 30)/(4 - 2)
Change in distance/Change in time = 30/2
Change in distance/Change in time = 15
Nathan’s rate of change is 15 miles per hour.
Reflect :
1. Recall that the graph of a proportional relationship is a line through the origin. Explain whether the relationship between Nathan’s time and distance is a proportional relationship.
Yes ; the graph is a line through the origin.
2. Does it matter what interval you use when you find the rate of change of a proportional relationship ? Explain.
No ; in a proportional relationship, the rate of change is constant.
Example 2 :
The graph shows the rate at which water is leaking from a tank. Find the rate at which the water is leaking from the tank per minute.
Solution :
Step 1 :
Identify the independent and dependent variables.
Independent : Time
Dependent : Leakage
Step 2 :
Find the rates of change.
0 hours to 4 hours :
Leakage/Change in time = (3 - 0)/(4 - 0)
Leakage/Change in time = 3/4
Leakage/Change in time = 0.75
4 hours to 8 hours :
Leakage/Change in time = (6 - 3)/(8 - 4)
Leakage/Change in time = 3/4
Leakage/Change in time = 0.75
1 hour to 8 hours :
Leakage/Change in time = (6 - 0)/(8 - 0)
Leakage/Change in time = 6/8
Leakage/Change in time = 3/4
Leakage/Change in time = 0.75
The water is leaking at the rate of 0.75 gallons per minute.
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