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]$h(t)=300-16 t^2$[/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]$A(\frac{3}{3})-A(\frac{0}{3})$[/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 for the average rate of change is 3 − 0 h ( 3 ) − h ( 0 ) .
Simplifying the expression, we get 3 h ( 3 ) − h ( 0 ) .
3 h ( 3 ) − h ( 0 )
Explanation
Understanding the Problem We are given the height function h ( t ) = 300 − 16 t 2 which models the height of a falling object t seconds after it is dropped from a platform 300 feet above the ground. We want to find the expression that represents 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 Comparing the simplified expression with the given options, we see that the correct expression is 3 h ( 3 ) − h ( 0 ) .
Examples
Understanding average rates of change is crucial in many real-world scenarios. For instance, consider a car accelerating from rest. The average acceleration over a certain time interval can be calculated using the same principle as the average rate of change. If the car's velocity at time t is given by v ( t ) , then the average acceleration between t 1 and t 2 is t 2 − t 1 v ( t 2 ) − v ( t 1 ) . This concept helps engineers design safer and more efficient vehicles.
The average rate of change of the height of the falling object from 0 to 3 seconds is calculated using the expression 3 h ( 3 ) − h ( 0 ) . This corresponds to option D in the multiple-choice question. The calculated value reflects a rate of − 48 feet per second, indicating the object is falling.
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