A package is dropped from a helicopter from a height of \( 3,136 \mathrm{ft} \). If the equation for height as a function of time is \( \mathrm{h}(\mathrm{t})=-16 \mathrm{t}^{2}+ \) initial height where t is time in seconds and \( \mathrm{h}(\mathrm{t}) \) is height in feet, how many seconds will it take for the package to hit the ground? \( [?] \) seconds

To find the time it takes for the package to hit the ground, we need to determine when the height of the package is zero. Given the equation for height as a function of time: \[ h(t) = -16t^2 + \text{initial height} \] Since the initial height is 3136 ft, we can substitute this value into the equation: \[ h(t) = -16t^2 + 3136 \] To find when the package hits the ground, we set the height to zero and solve for time: \[ 0 = -16t^2 + 3136 \] Now, we can solve this quadratic equation to find the time it takes for the package to hit the ground. Solve the quadratic equation by following steps: - step0: Solve using square roots: \(0=-16t^{2}+3136\) - step1: Swap the sides: \(-16t^{2}+3136=0\) - step2: Add or subtract both sides: \(-16t^{2}=-3136\) - step3: Divide both sides: \(\frac{-16t^{2}}{-16}=\frac{-3136}{-16}\) - step4: Divide the numbers: \(t^{2}=196\) - step5: Simplify the expression: \(t=\pm \sqrt{196}\) - step6: Simplify the expression: \(t=\pm 14\) - step7: Separate into possible cases: \(\begin{align}&t=14\\&t=-14\end{align}\) - step8: Rewrite: \(t_{1}=-14,t_{2}=14\) The solutions to the quadratic equation are \( t_1 = -14 \) and \( t_2 = 14 \). Since time cannot be negative, the package will hit the ground after 14 seconds.