If an object is propelled off a cliff, its height above the ground after [tex]$t$[/tex] seconds is given by the equation [tex]$s(t)=-9.8 t^2+148.6 t+30$[/tex], measured in meters. What is the average rate of change of the position of the object for the time period [tex]$t=1$[/tex] to [tex]$t=5$[/tex] seconds? The average rate of change of the object is $\square$ meters per second.
Answer (1)
Calculate the position at t = 1 : s ( 1 ) = − 9.8 ( 1 ) 2 + 148.6 ( 1 ) + 30 = 168.8 .
Calculate the position at t = 5 : s ( 5 ) = − 9.8 ( 5 ) 2 + 148.6 ( 5 ) + 30 = 528 .
Calculate the change in position: Δ s = s ( 5 ) − s ( 1 ) = 528 − 168.8 = 359.2 .
Calculate the average rate of change: Δ t Δ s = 5 − 1 359.2 = 89.8 . The average rate of change of the object is 89.8 meters per second.
Explanation
Problem Setup We are given the position function s ( t ) = − 9.8 t 2 + 148.6 t + 30 which describes the height of an object above the ground at time t . We want to find the average rate of change of the position of the object from t = 1 to t = 5 seconds. The average rate of change is given by the formula: Average rate of change = 5 − 1 s ( 5 ) − s ( 1 ) So, we need to calculate s ( 5 ) and s ( 1 ) .
Calculating s(1) First, let's calculate s ( 1 ) :
s ( 1 ) = − 9.8 ( 1 ) 2 + 148.6 ( 1 ) + 30 = − 9.8 + 148.6 + 30 = 168.8 So, s ( 1 ) = 168.8 meters.
Calculating s(5) Next, let's calculate s ( 5 ) :
s ( 5 ) = − 9.8 ( 5 ) 2 + 148.6 ( 5 ) + 30 = − 9.8 ( 25 ) + 743 + 30 = − 245 + 743 + 30 = 528 So, s ( 5 ) = 528 meters.
Calculating Average Rate of Change Now, we can calculate the average rate of change: Average rate of change = 5 − 1 s ( 5 ) − s ( 1 ) = 5 − 1 528 − 168.8 = 4 359.2 = 89.8 Therefore, the average rate of change of the position of the object from t = 1 to t = 5 seconds is 89.8 meters per second.
Final Answer The average rate of change of the position of the object for the time period t = 1 to t = 5 seconds is 89.8 meters per second.
Examples
Understanding average rates of change is crucial in many real-world scenarios. For example, consider a car accelerating from rest. The average rate of change of its velocity over a certain time interval tells us the average acceleration of the car during that period. Similarly, in economics, the average rate of change of a stock's price over a week can give investors an idea of the stock's performance. These concepts help us make informed decisions based on how quantities change over time.
