Question

question a potato is launched vertically upward with an initial velocity of 34 ft/s from a potato gun at the top of a building that is 42 feet tall. the distance, in feet, that the potato travels after t seconds is given by s(t)= - 16t^2 + 34t + 42. determine how long the potato is in the air. (enter an exact answer.) provide your answer below: seconds

Explanation:

Step1: Set the height function equal to 0

The height - function of the potato is $s(t)=-16t^{2}+34t + 42$. When the potato hits the ground, $s(t)=0$. So we have the quadratic equation $-16t^{2}+34t + 42 = 0$. Divide through by -2 to simplify: $8t^{2}-17t - 21=0$.

Step2: Use the quadratic formula

The quadratic formula for a quadratic equation $ax^{2}+bx + c = 0$ is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For the equation $8t^{2}-17t - 21 = 0$, where $a = 8$, $b=-17$, and $c=-21$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-17)^{2}-4\times8\times(-21)=289 + 672=961$. Then, $t=\frac{17\pm\sqrt{961}}{16}=\frac{17\pm31}{16}$.

Step3: Find the valid solution

We get two solutions for $t$: $t_1=\frac{17 + 31}{16}=\frac{48}{16}=3$ and $t_2=\frac{17-31}{16}=\frac{-14}{16}=-\frac{7}{8}$. Since time $t$ cannot be negative in this context, we discard the negative solution.

Answer:

3