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Problem Breakdown

The position function is given as:
$s(t) = -16t^2 + 104t$

The average velocity over a time interval $[t_1, t_2]$ is calculated as: $\text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

We will compute $s(t)$ for the given intervals and substitute into the formula.

Steps to Solve

1. Calculate $s(t_1)$ and $s(t_2)$ for each interval.
   Substitute $t_1 = 1$ and $t_2 = 2, 1.5, 1.1, 1.01, 1.001$ into $s(t)$.

2. Compute the average velocity.
   Use the formula for average velocity above.

Step-by-Step Calculation

1. $s(t_1 = 1)$: $s(1) = -16(1)^2 + 104(1) = -16 + 104 = 88$

2. For each $t_2$:

   • $t_2 = 2$:
     $s(2) = -16(2)^2 + 104(2) = -64 + 208 = 144$ Average velocity: $\frac{144 - 88}{2 - 1} = \frac{56}{1} = 56$

   • $t_2 = 1.5$:
     $s(1.5) = -16(1.5)^2 + 104(1.5) = -36 + 156 = 120$ Average velocity: $\frac{120 - 88}{1.5 - 1} = \frac{32}{0.5} = 64$

   • $t_2 = 1.1$:
     $s(1.1) = -16(1.1)^2 + 104(1.1) = -19.36 + 114.4 = 95.04$ Average velocity: $\frac{95.04 - 88}{1.1 - 1} = \frac{7.04}{0.1} = 70.4$

   • $t_2 = 1.01$:
     $s(1.01) = -16(1.01)^2 + 104(1.01) = -16.3232 + 105.04 = 88.7168$ Average velocity: $\frac{88.7168 - 88}{1.01 - 1} = \frac{0.7168}{0.01} = 71.68$

   • $t_2 = 1.001$:
     $s(1.001) = -16(1.001)^2 + 104(1.001) = -16.032016 + 104.104 = 88.071984$ Average velocity: $\frac{88.071984 - 88}{1.001 - 1} = \frac{0.071984}{0.001} = 71.984$

Completed Table

Conjecture:

As $t_2$ gets closer to $t_1 = 1$, the average velocity approaches 72. This suggests the instantaneous velocity at $t = 1$ is likely 72.

Would you like me to explain how this relates to derivatives or check any steps in more detail? 😊

Follow-Up Questions:

1. What does the instantaneous velocity represent in this context?
2. How does the average velocity formula relate to the derivative?
3. Can you derive $s'(t)$ to confirm the conjecture?
4. What happens if the position function includes higher powers of $t$?
5. How would the results change if $t_1$ were a different value?

Tip:

The instantaneous velocity is found by taking the derivative of the position function $s(t)$ and evaluating it at the given time $t$.