Suppose Doug kicks a soccer ball straight up into the air with an intial velocity of 96 feet per second.  The function mc006-1.jpg gives the height, in feet, of the soccer ball after t seconds?    What is the maximum height of the soccer ball?  (hint: Find the axis os symmetry and subsitute that answer back in.)

mc006-2.jpg

a
150 feet

b
6 feet

c
144 feet

d
136 feet

To solve this problem, let's first outline the steps based on the given hint. 

### 1. Understanding the Function:
The height \( h(t) \) of the soccer ball after \( t \) seconds can be given by a quadratic function, typically of the form:

\[
h(t) = -16t^2 + vt + s
\]

where:
- \( -16t^2 \) represents the acceleration due to gravity (in feet per second squared),
- \( v \) is the initial velocity (here, \( v = 96 \) feet per second),
- \( s \) is the initial height (which is usually 0 if the ball is kicked from ground level).

So, the function becomes:

\[
h(t) = -16t^2 + 96t
\]

### 2. Finding the Axis of Symmetry:
The axis of symmetry for a parabola given by \( ax^2 + bx + c \) is calculated using:

\[
t = -\frac{b}{2a}
\]

Here, \( a = -16 \) and \( b = 96 \).

\[
t = -\frac{96}{2(-16)} = \frac{96}{32} = 3 \text{ seconds}
\]

So, the time at which the maximum height is reached is \( t = 3 \) seconds.

### 3. Finding the Maximum Height:
Now, substitute \( t = 3 \) back into the height function \( h(t) \):

\[
h(3) = -16(3)^2 + 96(3)
\]

\[
h(3) = -16(9) + 288
\]

\[
h(3) = -144 + 288 = 144 \text{ feet}
\]

### 4. Answer:
The maximum height of the soccer ball is **144 feet**.

Thus, the correct answer is:

**c) 144 feet**

Would you like further details or have any questions?

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### Further Exploration:
1. What would be the height of the soccer ball after 1 second?
2. How long does it take for the soccer ball to hit the ground?
3. What is the initial height of the soccer ball?
4. How would the height function change if the ball were kicked from a 10-feet elevation?
5. What would be the maximum height if the initial velocity were 80 feet per second?
6. How would air resistance affect the maximum height of the soccer ball?
7. What is the velocity of the ball at its maximum height?
8. How does changing the initial velocity affect the time to reach the maximum height?

### Tip:
When analyzing projectile motion problems, always check if the problem assumes a vacuum (no air resistance) unless otherwise stated. This assumption simplifies calculations.

Learn how to find the maximum height of a soccer ball kicked with an initial velocity of 96 feet per second. This problem involves quadratic equations and projectile motion concepts. Suitable for high school students.

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