The height, [tex]$h$[/tex], of a falling object [tex]$t$[/tex] seconds after it is dropped from a platform 300 feet above the ground is modeled by the function [tex]$h(f)-300-16 t^2$[/tex]. Which expression could be used to determine the average rate at which the object falls during the first 3 seconds of its fall?
A. [tex]$h(3)-h(0)$[/tex]
B. [tex]$h(\frac{3}{3})-h(\frac{0}{3})$[/tex]
C. [tex]$\frac{h(3)}{3}$[/tex]
D. [tex]$\frac{h(3)-h(0)}{3}$[/tex]
Answer (2)
The average rate of change of a function h ( t ) over the interval [ a , b ] is given by b − a h ( b ) − h ( a ) .
We want to find the average rate of change of h ( t ) over the interval [ 0 , 3 ] .
The expression to evaluate is 3 − 0 h ( 3 ) − h ( 0 ) .
Simplifying the expression, we get 3 h ( 3 ) − h ( 0 ) .
The expression that represents the average rate at which the object falls during the first 3 seconds is 3 h ( 3 ) − h ( 0 ) .
Explanation
Understanding the Problem We are given the height function h ( t ) = 300 − 16 t 2 and we want to find the average rate at which the object falls during the first 3 seconds.
Average Rate of Change The average rate of change of a function h ( t ) over the interval [ a , b ] is given by b − a h ( b ) − h ( a ) . In this case, we want to find the average rate of change of h ( t ) over the interval [ 0 , 3 ] .
Evaluating the Expression So, we need to evaluate the expression 3 − 0 h ( 3 ) − h ( 0 ) .
Simplifying the Expression Simplifying the expression, we get 3 h ( 3 ) − h ( 0 ) .
Final Answer Therefore, the expression 3 h ( 3 ) − h ( 0 ) represents the average rate at which the object falls during the first 3 seconds.
Examples
Imagine you're tracking the descent of a drone from a certain height. The average rate of fall helps you understand how quickly the drone is losing altitude over a specific time period. This is crucial for monitoring its battery usage and ensuring it lands safely. By calculating this rate, you can make informed decisions about adjusting the drone's flight path or speed to optimize its performance and prevent any potential accidents.
To determine the average rate at which the object falls during the first 3 seconds, we calculate the average rate of change using the formula 3 h ( 3 ) − h ( 0 ) . This matches option D from the choices provided. The average rate at which the object falls is -48 feet per second.
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