Let's analyze the uploaded problem set. It focuses on the concepts of average velocity and its calculations using different formulas. I'll solve each of the three questions from the image for you:

1. Use the average velocity formula

The formula provided is:
$AV_{[a,b]} = \frac{s(b) - s(a)}{b - a}$ The function given is: $s(t) = -16t^2 + 5t + 7$, and we are calculating average velocity from $t = 0$ to $t = 2$.

Solution:

1. Compute $s(2)$:
   $s(2) = -16(2)^2 + 5(2) + 7 = -64 + 10 + 7 = -47$

2. Compute $s(0)$:
   $s(0) = -16(0)^2 + 5(0) + 7 = 7$

3. Substitute into the formula:
   $AV_{[0,2]} = \frac{s(2) - s(0)}{2 - 0} = \frac{-47 - 7}{2} = \frac{-54}{2} = -27 \, \text{ft/s}.$

2. Driving problem: Compute average velocity for a round trip

Details:

• Travel 120 miles east in 3 hours, so $v_1 = 120 / 3 = 40 \, \text{mph}$.
• Then drive 60 miles west in 1 hour, so $v_2 = -60 / 1 = -60 \, \text{mph}$.

Total displacement:

$\text{Displacement} = 120 - 60 = 60 \, \text{miles}.$

Total time:

$\text{Time} = 3 + 1 = 4 \, \text{hours}.$

Average velocity:

$AV = \frac{\text{Total displacement}}{\text{Total time}} = \frac{60}{4} = 15 \, \text{mph}.$

Correct answer: (b) 15 mph.

3. Use the alternate definition of average velocity

The formula is:
$AV_{[a, a+h]} = \frac{s(a + h) - s(a)}{h}$
The function is $s(t) = 64 - 16(t - 1)^2$, where $a = 0.8$, $h = 1.2 - 0.8 = 0.4$.

Solution:

1. Compute $s(0.8)$:
   $s(0.8) = 64 - 16(0.8 - 1)^2 = 64 - 16(-0.2)^2 = 64 - 16(0.04) = 64 - 0.64 = 63.36.$

2. Compute $s(1.2)$:
   $s(1.2) = 64 - 16(1.2 - 1)^2 = 64 - 16(0.2)^2 = 64 - 16(0.04) = 64 - 0.64 = 63.36.$

3. Compute average velocity:
   $AV_{[0.8, 1.2]} = \frac{s(1.2) - s(0.8)}{1.2 - 0.8} = \frac{63.36 - 63.36}{0.4} = \frac{0}{0.4} = 0 \, \text{ft/s}.$

Let me know if you need more clarification on these solutions!

Here are 5 related questions for practice:

1. What is the difference between velocity and speed?
2. How would the average velocity change in Question 2 if the return trip was faster or slower?
3. How do you interpret a negative average velocity?
4. If a particle's displacement is zero, what is its average velocity over that time interval?
5. What is the relationship between instantaneous velocity and average velocity?

Tip: Always keep track of direction when calculating velocity to ensure proper interpretation of positive and negative values.