https://file.aiquickdraw.com/mathbot/file/1726194000041mfo42kss.jpg

The problem in the image is asking to find the average velocity of an object over different intervals. The position of the object is given by the function:

\[ s(t) = -14t^2 + 42t \]

### Step 1: Formula for Average Velocity
The average velocity \( v_{\text{avg}} \) over a time interval \([a, b]\) is given by:

\[
v_{\text{avg}} = \frac{s(b) - s(a)}{b - a}
\]

### Step 2: Solve for the interval \([1, 7]\)
- First, compute the position \( s(1) \) and \( s(7) \).

\[
s(t) = -14t^2 + 42t
\]
- \( s(1) = -14(1)^2 + 42(1) = -14 + 42 = 28 \)
- \( s(7) = -14(7)^2 + 42(7) = -14(49) + 42(7) = -686 + 294 = -392 \)

### Step 3: Find the average velocity
\[
v_{\text{avg}} = \frac{s(7) - s(1)}{7 - 1} = \frac{-392 - 28}{6} = \frac{-420}{6} = -70
\]

### Final Answer:
The average velocity of the object over the interval \([1, 7]\) is \(-70\).

Would you like more details or have any questions?

Here are five related questions you might want to explore:
1. How do you compute the average velocity over the interval \([1, 6]\)?
2. How do you compute the average velocity over the interval \([1, 5]\)?
3. What is the physical significance of a negative average velocity?
4. How would you compute the instantaneous velocity at \( t = 3 \)?
5. How would the average velocity change if the position function was modified?

**Tip:** The average velocity gives a broad sense of how fast something is moving over an interval, but instantaneous velocity gives more precise, moment-by-moment information.

Find the average velocity of an object moving along a line given by the function s(t) = -14t^2 + 42t over various intervals.

Learn how to calculate the average velocity of an object using the position function s(t) = -14t^2 + 42t over different intervals. This guide provides a step-by-step solution using the average velocity formula and covers key concepts in calculus and quadratic functions. Suitable for students in Grades 9-12.

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