Question

suppose a firework is launched from a mortar with an upward velocity (initial speed) of about 112 ft/sec. the height h(t), in feet, of a weeping willow display, t seconds after having been launched from a 128 - foot high rooftop, is given by h(t)=-16t² + 112t + 128. how long will it take the cardboard shell from the fireworks to reach the ground? it will take the cardboard shell □ seconds to reach the ground. (simplify your answer.)

Explanation:

Step1: Set up the equation

When the cardboard shell reaches the ground, \(h(t)=0\). So we set \(-16t^{2}+112t + 128=0\). Divide the entire equation by \(-16\) to simplify: \(t^{2}-7t - 8=0\).

Step2: Factor the quadratic equation

We factor \(t^{2}-7t - 8\). We need two numbers \(a\) and \(b\) such that \(a + b=-7\) and \(ab=-8\). The numbers are \(-8\) and \(1\). So \(t^{2}-7t - 8=(t - 8)(t+1)=0\).

Step3: Solve for \(t\)

Using the zero - product property \(t - 8=0\) or \(t + 1=0\). Solving \(t - 8=0\) gives \(t = 8\), and solving \(t+1=0\) gives \(t=-1\). Since time \(t\geq0\), we discard \(t=-1\).

Answer:

\(8\)