To determine the average rate at which the object falls during the first 3 seconds of its fall, we need to calculate the average rate of change of the height function [tex]\( h(t) \)[/tex] over the interval from [tex]\( t = 0 \)[/tex] to [tex]\( t = 3 \)[/tex].
The height of the object at any time [tex]\( t \)[/tex] seconds can be expressed as:
[tex]\[ h(t) = 300 - 16t^2 \][/tex]
The heights at specific times are:
1. At [tex]\( t = 0 \)[/tex]:
[tex]\[ h(0) = 300 - 16 \cdot 0^2 = 300 \text{ feet} \][/tex]
2. At [tex]\( t = 3 \)[/tex]:
[tex]\[ h(3) = 300 - 16 \cdot 3^2 = 300 - 144 = 156 \text{ feet} \][/tex]
We know the object starts from a height of 300 feet and drops to 156 feet after 3 seconds.
The average rate of fall over this time interval is given by the formula for the average rate of change of the height function:
[tex]\[ \text{Average rate of fall} = \frac{h(3) - h(0)}{3 - 0} \][/tex]
Substituting the values we have:
[tex]\[ h(3) = 156 \][/tex]
[tex]\[ h(0) = 300 \][/tex]
Then,
[tex]\[ \text{Average rate of fall} = \frac{156 - 300}{3 - 0} = \frac{-144}{3} = -48 \text{ feet per second} \][/tex]
Therefore, the correct expression to determine the average rate at which the object falls during the first 3 seconds is:
[tex]\[ \frac{h(3) - h(0)}{3} \][/tex]
