The height in feet, [tex]$h$[/tex], of a model rocket [tex]$t$[/tex] seconds after launch is given by the equation [tex]$h(f)=3+70 t-16 t^2$[/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 (2)
The average rate of change of 6 feet per second indicates that the rocket ascends at this rate between t = 1 and t = 3 seconds. This means that the rocket's height increases consistently over this time period. The correct answer is option D .
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Calculate the height of the rocket at t = 1 and t = 3 : h ( 1 ) = 57 feet and h ( 3 ) = 69 feet.
Calculate the average rate of change between t = 1 and t = 3 : 3 − 1 h ( 3 ) − h ( 1 ) = 6 feet per second.
Interpret the average rate of change: The rocket's height changes at a constant rate of 6 feet per second between t = 1 and t = 3 .
The average rate of change tells us that the rocket is traveling at a constant rate of 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 told that the average rate of change of h ( t ) between t = 1 second and t = 3 seconds is 6. Our goal is to interpret what this average rate of change tells us about the rocket's motion.
Calculating Heights at t=1 and t=3 First, let's calculate the height of the rocket at t = 1 second and t = 3 seconds. At t = 1 , we have: h ( 1 ) = 3 + 70 ( 1 ) − 16 ( 1 ) 2 = 3 + 70 − 16 = 57 So, the height of the rocket at t = 1 second is 57 feet. At t = 3 , we have: h ( 3 ) = 3 + 70 ( 3 ) − 16 ( 3 ) 2 = 3 + 210 − 16 ( 9 ) = 3 + 210 − 144 = 69 So, the height of the rocket at t = 3 seconds is 69 feet.
Verifying the Average Rate of Change Now, let's calculate the average rate of change between t = 1 and t = 3 :
3 − 1 h ( 3 ) − h ( 1 ) = 3 − 1 69 − 57 = 2 12 = 6 This confirms that the average rate of change is indeed 6 feet per second.
Interpreting the Average Rate of Change The average rate of change represents the constant rate at which the rocket's height would need to change to have the same 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 during this time interval.
Comparing with Given Options Now, let's analyze the given options:
The rocket is traveling six times as fast when t = 3 than it is when t = 1 . This is not necessarily true. The average rate of change doesn't tell us about the instantaneous speeds at t = 1 and t = 3 , only the average speed over the interval.
The rocket is at a greater height when t = 3 than it is when t = 1 . This is true, since h ( 3 ) = 69 and h ( 1 ) = 57 . However, this is not what the average rate of change tells us. The average rate of change tells us about the change in height.
The rocket is 6 feet higher above the ground when t = 3 than it is when t = 1 . This is true since h ( 3 ) − h ( 1 ) = 69 − 57 = 12 , not 6. Also, this is not what the average rate of change tells us directly.
The rocket is traveling at a constant rate of 6 feet per second between t = 1 and t = 3 . This is the correct interpretation of the average rate of change.
Final Answer Therefore, the average rate of change of 6 feet per second tells us that the rocket is traveling at a constant rate of 6 feet per 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 stock price of a company over a period of time. The average rate of change tells you the average increase or decrease in the stock price per day. Similarly, in environmental science, you might track the average rate of change in temperature to understand climate trends. In physics, it helps in understanding the average velocity of an object over a certain time interval. These applications highlight how average rates of change provide valuable insights into trends and behaviors across various fields.
