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The problem in Task 2 involves analyzing the height function of a flying squirrel after it jumps, given by:

\[
h = -2t^2 + 9t + 1
\]

where \( h \) is the height in feet and \( t \) is the time in seconds. Let's go through each question:

### A. What was the squirrel's maximum height?
The height function \( h(t) = -2t^2 + 9t + 1 \) is a quadratic equation that opens downward (since the coefficient of \( t^2 \) is negative). The maximum height occurs at the vertex of this parabola.

The time at which the maximum height occurs can be found using the vertex formula:
\[
t = -\frac{b}{2a}
\]
Here, \( a = -2 \) and \( b = 9 \), so:
\[
t = -\frac{9}{2 \times -2} = \frac{9}{4} = 2.25 \text{ seconds}
\]

Now, substitute \( t = 2.25 \) into the height equation to find \( h \):
\[
h(2.25) = -2(2.25)^2 + 9(2.25) + 1
\]
\[
= -2(5.0625) + 20.25 + 1
\]
\[
= -10.125 + 20.25 + 1 = 11.125 \text{ feet}
\]

**Answer for A:** The squirrel's maximum height was **11.125 feet**.

### B. How many seconds after he jumped was the squirrel at his highest point?
From part A, we determined that the squirrel reached its maximum height at **2.25 seconds**.

**Answer for B:** **2.25 seconds**.

### C. When did the squirrel reach the ground?
The squirrel reaches the ground when \( h = 0 \). We need to solve for \( t \) in the equation:
\[
-2t^2 + 9t + 1 = 0
\]
Using the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = -2 \), \( b = 9 \), and \( c = 1 \).

Calculate the discriminant:
\[
b^2 - 4ac = 9^2 - 4(-2)(1) = 81 + 8 = 89
\]

Now, substitute into the formula:
\[
t = \frac{-9 \pm \sqrt{89}}{-4}
\]

Calculate each solution:
1. \( t = \frac{-9 + \sqrt{89}}{-4} \approx 4.69 \)
2. \( t = \frac{-9 - \sqrt{89}}{-4} \approx -0.11 \) (we discard this negative value as it doesn’t make sense in this context)

**Answer for C:** The squirrel reached the ground after approximately **4.69 seconds**.

---

Would you like more detailed steps, or do you have any questions about this solution?

Here are some related questions to consider:
1. How would the height function change if the squirrel jumped with a different initial velocity?
2. What would happen to the time of maximum height if the initial velocity was decreased?
3. How can we interpret the value of the discriminant in the context of the squirrel's jump?
4. How would the problem change if the squirrel jumped from a different initial height?
5. What would be the effect on the graph of \( h(t) \) if the coefficient of \( t^2 \) was different?

**Tip:** When solving quadratic equations in physics, checking the discriminant can give insight into the nature of the solutions before calculating them.

A flying squirrel jumped from a tree 1 feet in the air at an initial velocity of 9 feet per second. The equation h = -2t^2 + 9t + 1 models his jump where h is height in feet and t is time in seconds. 

A. What was the squirrel's maximum height?
B. How many seconds after he jumped was the squirrel at his highest point?
C. When did the squirrel reach the ground?

Analyze the height of a flying squirrel's jump with the quadratic equation h = -2t^2 + 9t + 1. Learn how to find the maximum height, the time at the highest point, and when the squirrel reaches the ground. Suitable for students in Grades 8-10, covering algebra and quadratic functions.

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