Question

question
a bullet is fired straight up from a bb gun with initial velocity 1,120 feet per second at an initial
height of 8 feet. use the formula ( h = - 16 t ^ { 2 } + v _ { 0 } t + 8 ) to determine how many seconds it
will take for the bullet to hit the ground. that is, when will ( h = 0 )? round your answer to the
nearest whole number.
provide your answer below:
( t = ) seconds
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Explanation:

Step1: Substitute \(v_0 = 1120\) and \(h = 0\) into the formula

We get the quadratic equation \(0=-16t^{2}+1120t + 8\). Divide through by \(-8\) to simplify: \(2t^{2}-140t - 1=0\). For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 2\), \(b=-140\), \(c = - 1\)), the quadratic formula is \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

Step2: Calculate the discriminant

\(\Delta=b^{2}-4ac=(-140)^{2}-4\times2\times(-1)=19600 + 8=19608\).

Step3: Apply the quadratic formula

\(t=\frac{140\pm\sqrt{19608}}{4}\). We have two solutions for \(t\): \(t_1=\frac{140+\sqrt{19608}}{4}\) and \(t_2=\frac{140-\sqrt{19608}}{4}\). Since time \(t>0\), we consider \(t=\frac{140+\sqrt{19608}}{4}\). \(\sqrt{19608}\approx140.03\), then \(t=\frac{140 + 140.03}{4}=\frac{280.03}{4}=70.0075\approx70\).

Answer:

\(t = 70\) seconds