ANCE ALGEBRA CFA- UNT 4 Spring $$ 2025 / 2 $$ of 10 The height, in feet, Lenny launched a t-shirt from a t-shirt cannon high in the stands at a football stadium is given by $$ h(x)=-16 x^{2}+64 x+80 $$, where $$ x $$ is the time in seconds after the $$ t $$-shirt is launched. Lenny was standing 16 feet height of 64

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To analyze the height of the t-shirt launched by Lenny, we have the quadratic function given by:

$$
h(x) = -16x^2 + 64x + 80
$$

where $$ h(x) $$ represents the height in feet, and $$ x $$ is the time in seconds after the t-shirt is launched.

### Step 1: Identify Known Conditions
- The function is a quadratic equation, which opens downwards (since the coefficient of $$ x^2 $$ is negative).
- The initial height of the t-shirt when $$ x = 0 $$ can be calculated by substituting $$ x = 0 $$ into the equation.

### Step 2: Calculate Initial Height
To find the initial height $$ h(0) $$:

$$
h(0) = -16(0)^2 + 64(0) + 80 = 80
$$

So, the t-shirt is launched from a height of 80 feet.

### Step 3: Find the Maximum Height
To find the maximum height of the t-shirt, we can use the vertex formula for a quadratic function, which occurs at $$ x = -\frac{b}{2a} $$. Here, $$ a = -16 $$ and $$ b = 64 $$.

Calculating the vertex:

$$
x = -\frac{64}{2 \cdot -16} = \frac{64}{32} = 2
$$

Now, we substitute $$ x = 2 $$ back into the height function to find the maximum height:

$$
h(2) = -16(2)^2 + 64(2) + 80
$$

Calculating this step-by-step:

1. Calculate $$ (2)^2 = 4 $$
2. Calculate $$ -16 \cdot 4 = -64 $$
3. Calculate $$ 64 \cdot 2 = 128 $$
4. Now substitute these values into the equation:

$$
h(2) = -64 + 128 + 80
$$

5. Combine the values:

$$
h(2) = 64 + 80 = 144
$$

Thus, the maximum height of the t-shirt is 144 feet.

### Step 4: Determine When the T-shirt Hits the Ground
To find when the t-shirt hits the ground, we need to solve for $$ h(x) = 0 $$:

$$
-16x^2 + 64x + 80 = 0
$$

We can use the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $$ a = -16 $$, $$ b = 64 $$, and $$ c = 80 $$.

Calculating the discriminant:

$$
b^2 - 4ac = 64^2 - 4(-16)(80)
$$

Calculating step-by-step:

1. Calculate $$ 64^2 = 4096 $$
2. Calculate $$ 4 \cdot -16 \cdot 80 = -5120 $$
3. Now substitute these values:

$$
4096 - (-5120) = 4096 + 5120 = 9216
$$

Now, applying the quadratic formula:

$$
x = \frac{-64 \pm \sqrt{9216}}{2 \cdot -16}
$$

Calculating $$ \sqrt{9216} $$:

$$
\sqrt{9216} = 96
$$

Now substituting back into the formula:

$$
x = \frac{-64 \pm 96}{-32}
$$

Calculating the two possible values for $$ x $$:

1. $$ x_1 = \frac{32}{-32} = -1 $$ (not valid since time cannot be negative)
2. $$ x_2 = \frac{-160}{-32} = 5 $$

Thus, the t-shirt hits the ground at $$ x = 5 $$ seconds.

### Summary of Results
- Initial height: 80 feet
- Maximum height: 144 feet at $$ x = 2 $$ seconds
- Time when the t-shirt hits the ground: 5 seconds

If you have any further questions or need additional calculations, feel free to ask!
The t-shirt was launched from an initial height of 80 feet. It reached a maximum height of 144 feet at 2 seconds and hit the ground after 5 seconds.

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