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

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!