Suppose an object is thrown upward with an initial velocity of 63 feet per second from a height of 105 feet. The height of the object [tex]$t$[/tex] seconds after it is thrown is given by [tex]$h(t)=-16 t^2+63 t+105$[/tex]. Find the average rate of change of the position of the object for the time period [tex]$t=0$[/tex] to [tex]$t=3$[/tex]. The average rate of change of the object is $\square$ $\square$.
Answer (1)
Calculate the height at t = 0 : h ( 0 ) = − 16 ( 0 ) 2 + 63 ( 0 ) + 105 = 105 .
Calculate the height at t = 3 : h ( 3 ) = − 16 ( 3 ) 2 + 63 ( 3 ) + 105 = 150 .
Calculate the change in height and time: Δ h = h ( 3 ) − h ( 0 ) = 150 − 105 = 45 and Δ t = 3 − 0 = 3 .
Calculate the average rate of change: Δ t Δ h = 3 45 = 15 .
Explanation
Problem Analysis and Setup We are given the height function h ( t ) = − 16 t 2 + 63 t + 105 , which describes the height of an object thrown upward with an initial velocity of 63 feet per second from a height of 105 feet. We want to find the average rate of change of the position of the object for the time period t = 0 to t = 3 . The average rate of change is given by the formula: 3 − 0 h ( 3 ) − h ( 0 ) So, we need to calculate h ( 0 ) and h ( 3 ) .
Calculating Initial Height First, let's calculate the height at t = 0 :
h ( 0 ) = − 16 ( 0 ) 2 + 63 ( 0 ) + 105 = 0 + 0 + 105 = 105 So, the initial height is 105 feet.
Calculating Height at t=3 Next, let's calculate the height at t = 3 :
h ( 3 ) = − 16 ( 3 ) 2 + 63 ( 3 ) + 105 = − 16 ( 9 ) + 189 + 105 = − 144 + 189 + 105 = 45 + 105 = 150 So, the height at t = 3 is 150 feet.
Calculating Average Rate of Change Now, we can calculate the average rate of change: 3 − 0 h ( 3 ) − h ( 0 ) = 3 150 − 105 = 3 45 = 15 The average rate of change of the position of the object for the time period t = 0 to t = 3 is 15 feet per second.
Final Answer Therefore, the average rate of change of the position of the object for the time period t = 0 to t = 3 is 15 feet per second.
Examples
Understanding average rates of change is crucial in many real-world scenarios. For instance, consider a car accelerating from rest. The average rate of change of its velocity over a certain time interval tells us how quickly the car's speed is increasing on average. This concept is also used in economics to analyze the growth rate of a company's revenue or the inflation rate of prices. By calculating these rates, we can make informed decisions and predictions about future trends.
