To determine the average rate at which the object falls during the first 3 seconds, we need to find the change in height over the change in time. This can be achieved by calculating the difference in height at [tex]\( t = 3 \)[/tex] seconds and [tex]\( t = 0 \)[/tex] seconds, and then dividing by the time interval, which is 3 seconds.
Given the height function:
[tex]\[ h(t) = 300 - 16t^2 \][/tex]
1. Calculate [tex]\( h(3) \)[/tex]:
[tex]\[ h(3) = 300 - 16(3)^2 = 300 - 144 = 156 \][/tex]
2. Calculate [tex]\( h(0) \)[/tex]:
[tex]\[ h(0) = 300 - 16(0)^2 = 300 \][/tex]
3. Determine the change in height over the 3-second interval:
[tex]\[ h(3) - h(0) = 156 - 300 = -144 \][/tex]
4. Divide the change in height by the time interval to find the average rate of fall:
[tex]\[ \frac{h(3) - h(0)}{3} = \frac{-144}{3} = -48 \][/tex]
Thus, the expression [tex]\(\frac{h(3) - h(0)}{3}\)[/tex] correctly determines the average rate at which the object falls during the first 3 seconds of its fall. Therefore, the answer is:
[tex]\[
\boxed{\frac{h(3) - h(0)}{3}}
\][/tex]
