If a rock is thrown upward on the planet Mars with a velocity 6 m/s, its height in meters t seconds later is given by y = 6t − 1.86t^2.
 (Round your answers to two decimal places.)
(a)
Find the average velocity (in m/s) over the given time intervals.

[1,1.1]____m/s

To calculate the average velocity of the rock over the time interval \([1, 1.1]\), we use the formula for average velocity:

\[
\text{Average Velocity} = \frac{y(t_2) - y(t_1)}{t_2 - t_1}
\]

where:
- \(y(t)\) is the height function \(y = 6t - 1.86t^2\),
- \(t_1 = 1\) and \(t_2 = 1.1\).

### Step 1: Calculate \(y(1)\)
Substitute \(t = 1\) into the height function:
\[
y(1) = 6(1) - 1.86(1)^2 = 6 - 1.86 = 4.14 \, \text{meters}
\]

### Step 2: Calculate \(y(1.1)\)
Substitute \(t = 1.1\) into the height function:
\[
y(1.1) = 6(1.1) - 1.86(1.1)^2
\]
First, calculate each term:
\[
6(1.1) = 6.6
\]
\[
1.1^2 = 1.21 \quad \text{so} \quad 1.86(1.21) = 2.2506
\]
\[
y(1.1) = 6.6 - 2.2506 = 4.3494 \, \text{meters}
\]

### Step 3: Calculate the average velocity
Now, use the average velocity formula:
\[
\text{Average Velocity} = \frac{y(1.1) - y(1)}{1.1 - 1} = \frac{4.3494 - 4.14}{0.1} = \frac{0.2094}{0.1} = 2.094 \, \text{m/s}
\]

### Answer:
The average velocity over the interval \([1, 1.1]\) is approximately **2.09 m/s**.

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Would you like further clarification, or do you have more questions?

Here are 5 related questions you might find interesting:
1. What is the average velocity over the interval \([1, 1.5]\)?
2. How can we find the instantaneous velocity at \(t = 1\)?
3. What is the maximum height the rock will reach?
4. How long does it take for the rock to hit the ground?
5. What is the initial acceleration of the rock?

**Tip:** Average velocity over a small interval can be a good approximation of instantaneous velocity at a point.

If a rock is thrown upward on the planet Mars with a velocity 6 m/s, its height in meters t seconds later is given by y = 6t − 1.86t^2. (Round your answers to two decimal places.) (a) Find the average velocity (in m/s) over the given time intervals [1,1.1].

Calculate the average velocity of a rock thrown upward on Mars using the quadratic height function y = 6t − 1.86t^2. This step-by-step solution explains how to find the average velocity over the time interval [1, 1.1] and covers important algebraic concepts. Suitable for high school students in Grades 9-12.

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