Is the rate of change constant? if so, what is the rate of change?

Question

During a hot-air balloon festival, data were gathered on the height (y) of a balloon after (x) minutes.

 

Hot air balloon height 

Time(min): 0, 2.2, 3.4, 4, 4.6

Height(m): 14, 80, 116, 134, 152

 

A. Is the rate of change constant? If so, what is the rate of change?

B. What are the units of the rate of change? What does the rate of change represent?

C. Write a linear equation in slope intercept form to model this situation. 

D. If the balloon continues to rise at the same rate, what will its height be after 8 minutes?

E. During what time interval is the height less than or equal to 500 meters ?

Answer 1

The rate of change is simply the slope between two points. In this case, it would be in m/min. Now, let's see if the rate of change is constant.

 

A)

 

m1=(80-14)/(2.2-0)=66/2.2=30 m/min

m2=(116-80)/(3.4-2.2)=36/1.2=30 m/min

m3=(134-116)/(4-3.4)=18/0.6=30 m/min

m4=152-134)/(4.6-4)=18/0.6=30 m/min

 

Thus, the rate of change is constant, and it is 30 m/min

 

B) The units of the rate of change is m/min. The rate of change represents the speed in which the balloon rises.

 

C) Lets start with writing the equation in point slope form at the point (0,14)

 

y-14=30(x-0)=30x  Hence, the equation is y=30x+14

 

D) y=30*8+14=240+14=254 m

 

E)   y=30x+14≤500

               -14   -14

                 30x≤486 Divide both sides of the equation by 30.

 

                   x≤16.67 min  Hence, the time interval in which the balloon is 500 meters or less is 0 and 16.67 minutes.