Question

a potato is launched vertically upward with an initial velocity of 32 ft/s from a potato gun at the top of a building that is 99 feet tall. the height of the potato in feet after t seconds is given by s(t)= - 16t² + 32t + 99. when is the potato at a height of 79 ft? enter an exact answer. provide your answer below: seconds

Explanation:

Step1: Set up the equation

Set $s(t)=79$, so $- 16t^{2}+32t + 99=79$.

Step2: Rearrange the equation

Subtract 79 from both sides to get $-16t^{2}+32t + 20 = 0$. Divide through by -4 to simplify: $4t^{2}-8t - 5=0$.

Step3: Use the quadratic formula

For a quadratic equation $ax^{2}+bx + c = 0$ ($a = 4$, $b=-8$, $c = - 5$), the quadratic formula is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute the values: $t=\frac{8\pm\sqrt{(-8)^{2}-4\times4\times(-5)}}{2\times4}$.

Step4: Simplify the expression inside the square - root

Calculate $(-8)^{2}-4\times4\times(-5)=64 + 80=144$. Then $t=\frac{8\pm\sqrt{144}}{8}=\frac{8\pm12}{8}$.

Step5: Find the two solutions for t

We have two cases: $t_1=\frac{8 + 12}{8}=\frac{20}{8}=\frac{5}{2}$ and $t_2=\frac{8-12}{8}=\frac{-4}{8}=-\frac{1}{2}$. Since time $t\geq0$, we discard the negative solution.

Answer:

$\frac{5}{2}$