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]$h = 300 - 16t^2$[/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]$4\left(\frac{3}{3}\right)-4\left(\frac{0}{3}\right)$[/tex]
C. [tex]$\frac{h(3)}{3}$[/tex]
D. [tex]$\frac{h(3)-h(0)}{3}$[/tex]
Answer (1)
The problem asks for the expression representing the average rate of change of a falling object's height over the first 3 seconds.
The average rate of change is calculated using the formula b − a h ( b ) − h ( a ) .
Applying this to the interval [ 0 , 3 ] , the expression becomes 3 − 0 h ( 3 ) − h ( 0 ) .
Simplifying, the expression is 3 h ( 3 ) − h ( 0 ) .
Explanation
Understanding the Problem We are given the height function h ( t ) = 300 − 18 t 2 and asked to find the expression that represents the average rate of change of the height of the object with respect to time during the first 3 seconds. The average rate of change of a function h ( t ) over the interval [ a , b ] is given by the formula b − a h ( b ) − h ( a ) . In this case, we want to find the average rate of change over the interval [ 0 , 3 ] . So, we need to calculate 3 − 0 h ( 3 ) − h ( 0 ) . This simplifies to 3 h ( 3 ) − h ( 0 ) .
Finding the Average Rate of Change The average rate of change of the height of the object with respect to time during the first 3 seconds is given by the expression 3 h ( 3 ) − h ( 0 ) .
Final Answer The expression that could be used to determine the average rate at which the object falls during the first 3 seconds of its fall is 3 h ( 3 ) − h ( 0 ) .
Examples
Understanding average rates of change is crucial in many real-world scenarios. For example, if you're tracking the distance a car travels over time, the average rate of change (or average speed) tells you how quickly the car is moving on average during that time period. Similarly, in finance, you might use average rates of change to analyze the growth of an investment over a certain period. This concept helps in making informed decisions based on how things change over time.
