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Let's analyze the problem step by step.

### Given:
The distance \( s(t) \) in feet of an object from a point is given by:
\[
s(t) = t^2
\]
where \( t \) is in seconds.

#### (a) Average Velocity between \( t = 6 \) and \( t = 8 \):

The average velocity is calculated by the formula:
\[
\text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]
where \( t_1 = 6 \) and \( t_2 = 8 \).

1. \( s(6) = 6^2 = 36 \)
2. \( s(8) = 8^2 = 64 \)

Now, using the formula:
\[
\text{Average Velocity} = \frac{64 - 36}{8 - 6} = \frac{28}{2} = 14 \, \text{ft/sec}
\]

So, the average velocity between \( t = 6 \) and \( t = 8 \) is **14 ft/sec**.

---

#### (b) Instantaneous Velocity at \( t = 6 \):

The instantaneous velocity is found by taking the derivative of \( s(t) \) with respect to \( t \). This gives us the velocity function \( v(t) \).

Since \( s(t) = t^2 \), the derivative is:
\[
v(t) = \frac{d}{dt} \left( t^2 \right) = 2t
\]

To find the instantaneous velocity at \( t = 6 \):
\[
v(6) = 2 \times 6 = 12 \, \text{ft/sec}
\]

Thus, the instantaneous velocity at \( t = 6 \) is **12 ft/sec**.

---

Let me know if you'd like further details or clarifications!

### Questions for Deeper Understanding:
1. What is the formula for calculating average velocity over an interval?
2. How do you interpret the concept of instantaneous velocity in a physical context?
3. Why is the derivative used to find instantaneous velocity?
4. How does the average velocity differ from the instantaneous velocity?
5. What are some practical applications of calculating instantaneous velocity in physics?

### Tip:
When calculating instantaneous velocity, always remember that it represents the slope of the tangent line to the distance-time curve at a given point, which is the same as the derivative of the distance function at that point.

The distance (in feet) of an object from a point is given by s(t) = t^2, where t is in seconds. (a) What is the average velocity of the object between t = 6 and t = 8? (b) By using smaller and smaller intervals around 6, estimate the instantaneous velocity at time t = 6. Round your answer to the nearest integer.

Calculate the average velocity between t = 6 and t = 8, and estimate the instantaneous velocity at t = 6 for the function s(t) = t^2. This problem involves key calculus concepts such as rate of change, differentiation, and velocity. Suitable for high school students in Grades 10-12.

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