To find the approximate distance the rock had fallen at 3.5 seconds, let's use the information provided in the tables and some basic principles of motion.
1. Current knowledge of distance fallen at certain times:
- At 3 seconds, the rock had fallen 144 feet.
- At 4 seconds, the rock had fallen 256 feet.
2. Velocities at known times:
- At 3 seconds, the velocity was 96 ft/s.
- At 4 seconds, the velocity was 128 ft/s.
3. Average Velocity to estimate distance over the next 0.5 seconds:
- To find the distance fallen between 3 and 4 seconds, we can calculate using average velocity. The average velocity between 3 seconds and 4 seconds is:
[tex]\[
\text{Average Velocity} = \frac{96 \, \text{ft/s} + 128 \, \text{ft/s}}{2} = 112 \, \text{ft/s}
\][/tex]
4. Determine the distance fallen in the next 0.5 seconds:
- Using the average velocity, the additional distance covered in the next 0.5 seconds can be approximated as:
[tex]\[
\text{Distance over 0.5 seconds} = 112 \, \text{ft/s} \times 0.5 \, \text{s} = 56 \, \text{ft}
\][/tex]
5. Total distance fallen at 3.5 seconds:
- The total distance the rock had fallen by 3.5 seconds is the sum of the distance fallen by 3 seconds and the additional distance fallen in the next 0.5 seconds:
[tex]\[
\text{Total Distance at 3.5 seconds} = 144 \, \text{ft} + 56 \, \text{ft} = 200 \, \text{ft}
\][/tex]
Therefore, the approximate distance the rock had fallen at 3.5 seconds is [tex]\( \boxed{200 \text{ ft}} \)[/tex].
