The height in feet, [tex]$h$[/tex], of a model rocket [tex]$t$[/tex] seconds after launch is given by the equation [tex]$h(t)=3+70 t-16 t^2$[/tex]. The average rate of change in [tex]$h(t)$[/tex] between [tex]$t=1$[/tex] second and [tex]$t=3$[/tex] second is 6. What does the average rate of change tell you about the rocket?
A. The rocket is traveling six times as fast when [tex]$t=3$[/tex] than it is when [tex]$t=1$[/tex].
B. The rocket is at a greater height when [tex]$t=3$[/tex] than it is when [tex]$t=1$[/tex].
C. The rocket is 6 feet higher above the ground when [tex]$t=3$[/tex] than it is when [tex]$t=1$[/tex].
D. The rocket is traveling at a constant rate of 6 feet per second between [tex]$t=1$[/tex] and [tex]$t=3$[/tex].
Answer (1)
Calculate the height at t = 1 : h ( 1 ) = 57 feet.
Calculate the height at t = 3 : h ( 3 ) = 69 feet.
The average rate of change is 3 − 1 h ( 3 ) − h ( 1 ) = 6 feet per second.
The rocket is at a greater height when t = 3 than when t = 1 , and the average rate of change is 6 feet per second between t = 1 and t = 3 .
Explanation
Understanding the Problem We are given the height of a model rocket as a function of time, h ( t ) = 3 + 70 t − 16 t 2 . We are also given that the average rate of change of h ( t ) between t = 1 second and t = 3 seconds is 6. We need to determine what this average rate of change tells us about the rocket's motion.
Calculating Heights First, let's calculate the height of the rocket at t = 1 and t = 3 seconds.
Height at t=1 At t = 1 second: h ( 1 ) = 3 + 70 ( 1 ) − 16 ( 1 ) 2 = 3 + 70 − 16 = 57 So, h ( 1 ) = 57 feet.
Height at t=3 At t = 3 seconds: h ( 3 ) = 3 + 70 ( 3 ) − 16 ( 3 ) 2 = 3 + 210 − 16 ( 9 ) = 3 + 210 − 144 = 69 So, h ( 3 ) = 69 feet.
Calculating Average Rate of Change Now, let's calculate the average rate of change between t = 1 and t = 3 :
Average rate of change = 3 − 1 h ( 3 ) − h ( 1 ) = 3 − 1 69 − 57 = 2 12 = 6 The average rate of change is indeed 6 feet per second, as given.
Interpreting Average Rate of Change The average rate of change represents the constant rate at which the rocket's height would have to change to achieve the same overall change in height between t = 1 and t = 3 . In other words, if the rocket were traveling at a constant speed, that speed would be 6 feet per second to cover the same height difference in the same time interval.
Evaluating Statements Now, let's evaluate the given statements:
The rocket is traveling six times as fast when t = 3 than it is when t = 1 . This statement is incorrect. The average rate of change does not tell us about the instantaneous speed of the rocket at t = 1 or t = 3 .
The rocket is at a greater height when t = 3 than it is when t = 1 . This statement is correct. We calculated h ( 1 ) = 57 and h ( 3 ) = 69 , so the rocket is indeed higher at t = 3 .
The rocket is 6 feet higher above the ground when t = 3 than it is when t = 1 . This statement is correct. The difference in height is h ( 3 ) − h ( 1 ) = 69 − 57 = 12 feet. The average rate of change is 6 feet per second, but the height difference is 12 feet.
The rocket is traveling at a constant rate of 6 feet per second between t = 1 and t = 3 . This statement is correct. The average rate of change is 6 feet per second, which means that on average , the rocket's height increases by 6 feet each second during this interval. It does not mean the rocket is traveling at a constant rate of 6 feet per second. The rocket's speed is changing due to gravity.
Final Answer Since h ( 3 ) = 69 and h ( 1 ) = 57 , the rocket is at a greater height when t = 3 than when t = 1 . The average rate of change of 6 feet per second tells us that the rocket's height increased by an average of 6 feet each second between t = 1 and t = 3 .
Examples
Understanding average rates of change is crucial in many real-world scenarios. For instance, imagine you're tracking the growth of a plant over time. If the plant grows 2 inches in the first week and 4 inches in the second week, the average growth rate is (4-2)/(2-1) = 2 inches per week. This tells you the typical growth per week over that period, even if the actual growth varied each week. Similarly, in economics, the average rate of change can represent the average inflation rate or the average growth rate of a company's revenue over a certain period, providing a simplified view of overall trends.
