Consider an object moving along a line with the given velocity . Assume is time measured in seconds and velocities have units of .
a. Determine when the motion is in the positive direction and when it is in the negative direction.
b. Find the displacement over the given interval.
c. Find the distance traveled over the given interval.

a. When is the motion in the positive direction? Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The motion is in the positive direction for -values in the interval(s)
(Use a comma to separate answers as needed. Type your answers in interval notation.)
B. The motion is never in the positive direction.

Ask by Ramirez Carrillo. in the United States

Jan 23,2025

Question

Consider an object moving along a line with the given velocity . Assume is time measured in seconds and velocities have units of .
a. Determine when the motion is in the positive direction and when it is in the negative direction.
b. Find the displacement over the given interval.
c. Find the distance traveled over the given interval.

a. When is the motion in the positive direction? Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A. The motion is in the positive direction for -values in the interval(s)
(Use a comma to separate answers as needed. Type your answers in interval notation.)
B. The motion is never in the positive direction.

Ask by Ramirez Carrillo. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To determine when the motion is in the **positive direction**, we need to analyze the velocity function:

$$
v(t) = 2t^2 - 16t + 30
$$

**Positive direction** corresponds to when the velocity $$ v(t) > 0 $$.

### Step 1: Find the Critical Points
Set $$ v(t) = 0 $$ to find the critical points:

$$
2t^2 - 16t + 30 = 0
$$

Divide the equation by 2:

$$
t^2 - 8t + 15 = 0
$$

Factor or use the quadratic formula:

$$
t = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm 2}{2}
$$

So,

$$
t = 5 \quad \text{and} \quad t = 3
$$

### Step 2: Determine Intervals Where $$ v(t) > 0 $$
Since the coefficient of $$ t^2 $$ is positive, the parabola opens upwards. Therefore, $$ v(t) > 0 $$ outside the roots.

For the interval $$[0, 6]$$:

- $$ v(t) > 0 $$ when $$ t < 3 $$ and $$ t > 5 $$.

### Conclusion
The motion is in the **positive direction** for $$ t $$-values in the intervals:

$$
(0, 3) \quad \text{and} \quad (5, 6)
$$

**Therefore, the correct choice is:**

**A.** The motion is in the positive direction for $$ t $$-values in the intervals $$(0, 3)$$ and $$(5, 6)$$.
The motion is in the positive direction for $$ t $$-values in the intervals $$(0, 3)$$ and $$(5, 6)$$.

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