To determine the average rate at which the object falls during the first 3 seconds, we need to analyze how the height [tex]\( h \)[/tex] of the object changes over this time period. Let's start by calculating the height at [tex]\( t = 3 \)[/tex] seconds and [tex]\( t = 0 \)[/tex] seconds using the function provided, [tex]\( h(t) = 300 - 16t^2 \)[/tex].
1. Calculate [tex]\( h(3) \)[/tex]:
[tex]\[
h(3) = 300 - 16 \cdot 3^2 = 300 - 144 = 156
\][/tex]
2. Calculate [tex]\( h(0) \)[/tex]:
[tex]\[
h(0) = 300 - 16 \cdot 0^2 = 300
\][/tex]
Next, we will evaluate the expressions given to find the average rate at which the object falls and compare them.
### Evaluating each option:
Option 1: [tex]\( h(3) - h(0) \)[/tex]:
[tex]\[
h(3) - h(0) = 156 - 300 = -144
\][/tex]
This indicates the change in height over the 3 seconds is -144 feet. However, it does not directly give the average rate of fall.
Option 2: [tex]\( h\left(\frac{3}{3}\right) - h\left(\frac{0}{3}\right) \)[/tex]:
[tex]\[
h(1) - h(0) = 300 - 16 \cdot 1^2 - 300 = 300 - 16 - 300 = -16
\][/tex]
This calculates the change in height over the first second ( [tex]\( h(1) - h(0) \)[/tex] ).
Option 3: [tex]\( \frac{h(3)}{3} \)[/tex]:
[tex]\[
\frac{h(3)}{3} = \frac{156}{3} = 52
\][/tex]
This gives the average height at [tex]\( t = 3 \)[/tex] seconds, not the average rate of fall.
Option 4: [tex]\( \frac{w(x) - w(y)}{3} \)[/tex]:
We'll ignore this option as it's not related to the given [tex]\( h(t) \)[/tex] function.
### Conclusion:
Among the given options, option 1 measures the change in height over the first 3 seconds, but it does not divided by the time duration to find the average rate of fall. It simply gives the net distance fallen.
Upon evaluating the options, the combination of measurements and interpretations leads us to recognize that:
- [tex]\( -144 \)[/tex] represents the total change in height.
- [tex]\( -16 \)[/tex] is the correct average change per second when calculating the change from [tex]\( t = 0 \)[/tex] to [tex]\( t = 1 \)[/tex] which is misleading for 3 seconds.
- And, [tex]\( 52 \)[/tex] is not valid average rate but average height.
Thus, taking a closer look, none of the options seem perfect as they missed the right calculation explained and context for average rate of fall interpretation and selective options here.
