Distance is calculated as the product of speed and time. If a vehicle traveled d(x)=6x2−x−2 in kilometers for t(x)=3x−2 hours, what is the car's average speed in terms of x?


To find the car's average speed in terms of \( x \), we use the formula for speed:

\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Given:
- Distance function \( d(x) = 6x^2 - x - 2 \) (in kilometers),
- Time function \( t(x) = 3x - 2 \) (in hours),

The average speed in terms of \( x \) is:

\[
\text{Average Speed} = \frac{d(x)}{t(x)} = \frac{6x^2 - x - 2}{3x - 2}
\]

Now, simplify the expression. To do so, we can attempt polynomial division.

### Division of \( 6x^2 - x - 2 \) by \( 3x - 2 \):

1. Divide the leading term \( 6x^2 \) by \( 3x \), which gives \( 2x \).
2. Multiply \( 2x \) by \( 3x - 2 \), which gives \( 6x^2 - 4x \).
3. Subtract \( (6x^2 - 4x) \) from \( 6x^2 - x - 2 \), which leaves \( 3x - 2 \).
4. Divide \( 3x \) by \( 3x \), which gives \( 1 \).
5. Multiply \( 1 \) by \( 3x - 2 \), which gives \( 3x - 2 \).
6. Subtract \( (3x - 2) \) from \( 3x - 2 \), which leaves 0.

Thus, the quotient is:

\[
\frac{6x^2 - x - 2}{3x - 2} = 2x + 1
\]

Therefore, the car's average speed in terms of \( x \) is:

\[
\boxed{2x + 1} \, \text{km/h}
\]

Would you like further details on polynomial division or any part of the process?

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Here are some related questions:

1. How can you calculate instantaneous speed from distance and time functions?
2. What happens to average speed if the time function changes?
3. How do you interpret the meaning of the average speed function \( 2x + 1 \)?
4. Can we solve this using limits for a more precise average speed?
5. What is the significance of simplifying rational functions in physics?

**Tip:** Always check if polynomials can be simplified before division, as it can make calculations easier!

Distance is calculated as the product of speed and time. If a vehicle traveled d(x) = 6x^2 − x − 2 in kilometers for t(x) = 3x − 2 hours, what is the car's average speed in terms of x?

Learn how to calculate the average speed of a vehicle given the distance function d(x) = 6x^2 − x − 2 and time function t(x) = 3x − 2 using polynomial division. This solution explains key algebraic concepts like rational functions and polynomial division. Suitable for Grades 9-12 students.

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