Let's break down the two problems you have shared:

35. Average velocity

The velocity $v(t)$ is given as $v(t) = t^2 + 3t$ over the time interval $[0, 6]$.

To find the average velocity:

The average velocity is given by the formula:

$\text{Average velocity} = \frac{1}{b-a} \int_{a}^{b} v(t) \, dt$

where $a = 0$, $b = 6$, and $v(t) = t^2 + 3t$.

So, the formula becomes:

$\text{Average velocity} = \frac{1}{6 - 0} \int_0^6 (t^2 + 3t) \, dt$

Let me calculate this for you.

36. Average velocity of a rock

The position $s(t)$ is given by the equation $s(t) = -16t^2 + 64$. You are asked to find the average velocity of the rock during its flight.

To find the average velocity:

We use the formula:

$\text{Average velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

where $s(t) = -16t^2 + 64$, $t_1$ is the time at launch, and $t_2$ is the time when the rock hits the ground.

1. At $t_1 = 0$, $s(0) = 64$ feet (the initial height).
2. To find $t_2$, set $s(t) = 0$ to determine when the rock hits the ground:

$-16t^2 + 64 = 0$

Let me solve both these steps for you.Here are the solutions for the problems:

Problem 35:

The average velocity of the object over the time interval $[0, 6]$ is: $\text{Average velocity} = 21 \, \text{m/s}$

Problem 36:

The average velocity of the rock during its flight (from launch to when it hits the ground) is: $\text{Average velocity} = -32 \, \text{ft/s}$ (The negative sign indicates that the average motion is downward.)

Would you like more details on any of these steps or have further questions?

Here are 5 related questions to expand your understanding:

1. How do you interpret the physical meaning of average velocity when it is negative?
2. How would the process change if you were given a non-quadratic velocity function?
3. Can you find the total distance traveled by the object in problem 35 using calculus?
4. What happens if we extend the time interval beyond 6 seconds for problem 35?
5. How would the velocity function change for the rock in problem 36 if air resistance was considered?

Tip: Always ensure to correctly identify the time intervals when calculating average velocity. The interpretation of the results depends heavily on the limits chosen!