Option C: It takes 3 seconds for the ball to hit the ground.
Explanation:
The given quadratic function is [tex]h=-16 t^{2}+46 t+6[/tex] where h is the height in feet and t is the time in seconds.
We need to determine at what time the ball will hit the ground.
Time taken:
The time can be determined by substituting h = 0 in the function [tex]h=-16 t^{2}+46 t+6[/tex]
Thus, we get;
[tex]0=-16 t^{2}+46 t+6[/tex]
Let us solve the quadratic expression using the quadratic formula.
Thus, we have;
[tex]t=\frac{-46 \pm \sqrt{46^{2}-4(-16) 6}}{2(-16)}[/tex]
Solving, we get,
[tex]t=\frac{-46 \pm \sqrt{2116+384}}{-32}[/tex]
[tex]t=\frac{-46 \pm \sqrt{2500}}{-32}[/tex]
[tex]t=\frac{-46 \pm 50}{-32}[/tex]
Thus, the values of t are given by
[tex]t=\frac{-46 + 50}{-32}[/tex] and [tex]t=\frac{-46 - 50}{-32}[/tex]
[tex]t=\frac{4}{-32}[/tex] and [tex]t=\frac{-96}{-32}[/tex]
[tex]t=-\frac{1}{8}[/tex] and [tex]t=3[/tex]
Since, t cannot take negative values.
Thus, the value of t is [tex]t=3[/tex]
Hence, the time taken by the ball to hit the ground is 3 seconds.
Therefore, Option C is the correct answer.
