We are given the height function of a projectile:

$h(t) = -16t^2 + 64t + 192$

(a) Finding the maximum height:

This is a quadratic function, and the maximum height occurs at the vertex. The time $t$ at which the vertex occurs can be found using the formula:

$t = \frac{-b}{2a}$

Where $a = -16$ and $b = 64$. Plugging in these values:

$t = \frac{-64}{2(-16)} = \frac{-64}{-32} = 2 \, \text{seconds}$

Now, substitute $t = 2$ into the height function to find the maximum height:

$h(2) = -16(2)^2 + 64(2) + 192 = -16(4) + 128 + 192 = -64 + 128 + 192 = 256 \, \text{feet}$

So, the maximum height is 256 feet.

(b) The time when the projectile achieves its maximum height:

From the previous calculation, the maximum height occurs at $t = 2$ seconds.

(c) Finding the time when the projectile has a height of 0 feet:

To find when the projectile reaches a height of 0 feet, we set $h(t) = 0$:

$-16t^2 + 64t + 192 = 0$

This is a quadratic equation. We can solve it using the quadratic formula:

$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = -16$, $b = 64$, and $c = 192$. Plugging these values into the quadratic formula:

$t = \frac{-64 \pm \sqrt{64^2 - 4(-16)(192)}}{2(-16)}$ $t = \frac{-64 \pm \sqrt{4096 + 12288}}{-32}$ $t = \frac{-64 \pm \sqrt{16384}}{-32}$ $t = \frac{-64 \pm 128}{-32}$

Now, solving for both possible values of $t$:

1. $t = \frac{-64 + 128}{-32} = \frac{64}{-32} = -2$ (This is not physically meaningful since time cannot be negative).
2. $t = \frac{-64 - 128}{-32} = \frac{-192}{-32} = 6$.

Thus, the time when the projectile hits the ground is 6 seconds.

Summary:

• (a) Maximum height = 256 feet
• (b) Time to reach maximum height = 2 seconds
• (c) Time when the projectile hits the ground = 6 seconds

Would you like more details on any part of this problem?

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Here are 5 related questions for further practice:

1. What would happen to the maximum height if the initial velocity were increased to 80 feet per second?
2. How would the equation change if the initial height were 100 feet?
3. Can you determine when the projectile reaches a height of 150 feet?
4. What is the total flight time of the projectile if the initial velocity was halved?
5. What is the velocity of the projectile at $t = 3$ seconds?

Tip: The quadratic formula can always help solve for time when height or position is zero!