Let's solve each part step-by-step.

Part (a)

To estimate $v'(8)$ using the data table, we need to approximate the derivative at $t = 8$. Since $v'(t)$ represents the rate of change of $v(t)$, we can use the values of $v(t)$ at $t = 6$ and $t = 10$ to estimate $v'(8)$ with the average rate of change formula:

$v'(8) \approx \frac{v(10) - v(6)}{10 - 6} = \frac{600 - 500}{10 - 6} = \frac{100}{4} = 25 \, \text{meters per minute squared}$

So, $v'(8) \approx 25 \, \text{m/min}^2$.

Part (b)

To find John's average acceleration over the interval $6 \leq t \leq 12$, we calculate the average rate of change of $v(t)$ between $t = 6$ and $t = 12$:

$\text{Average acceleration} = \frac{v(12) - v(6)}{12 - 6} = \frac{-430 - 500}{12 - 6} = \frac{-930}{6} = -155 \, \text{m/min}^2$

So, the average acceleration over this interval is $-155 \, \text{m/min}^2$.

Part (c)

Sam’s velocity is given by $v(t) = 0.5t^3 - 2t^2 + 100$. To find her acceleration at $t = 3$, we need to find $v'(t)$, the derivative of $v(t)$, and then evaluate it at $t = 3$.

1. Differentiate $v(t)$: $v'(t) = \frac{d}{dt} (0.5t^3 - 2t^2 + 100) = 1.5t^2 - 4t$

2. Substitute $t = 3$: $v'(3) = 1.5(3)^2 - 4(3) = 1.5 \cdot 9 - 12 = 13.5 - 12 = 1.5 \, \text{m/min}^2$

So, Sam’s acceleration at $t = 3$ is $1.5 \, \text{m/min}^2$.

Part (d)

At $t = 3$, John is at $x_J = 200$ meters, and Sam is at $x_S = 300$ meters. We are asked to find the rate at which the distance between John and Sam is changing at $t = 3$.

1. The distance between John and Sam at any time $t$ is $D(t) = |x_S(t) - x_J(t)|$.
2. Since we are given only their velocities, we will interpret $v_J(3)$ and $v_S(3)$ as the rates of change of their positions with respect to time.

Using $v_J(3) = 500$ m/min (from the table) and $v_S(3) = v(3) = 0.5(3)^3 - 2(3)^2 + 100 = 94.5$ m/min:

$\frac{dD}{dt} = v_S(3) - v_J(3) = 94.5 - 500 = -405.5 \, \text{m/min}$

Thus, the distance between John and Sam is decreasing at a rate of $405.5 \, \text{m/min}$ at $t = 3$.

Would you like further explanations on any part? Here are related questions for practice:

1. How would you estimate $v'(t)$ if more data points were available around $t = 8$?
2. What would be John's position at $t = 10$ if his initial position was 0?
3. How can we determine Sam's position function if we know her velocity function?
4. What is the significance of a negative acceleration in the context of John's motion?
5. How would the distance between John and Sam change if Sam had a different initial velocity?

Tip: For average rates of change, remember to focus on initial and final values within the interval!