Answer:
2, 5
Step-by-step explanation:
We are given the position function of a ball (in feet):
[tex]s(t) = 112t - 16t^2[/tex]
We are solving for the time when the ball is 160 feet from its starting position (the ground).
We can get this by setting s(t) to 160 and solving for t:
[tex]160 = 112t-16t^2[/tex]
↓ getting all the terms on one side
[tex]16t^2 - 112t + 160=0[/tex]
↓ dividing both sides by 16
[tex]t^2 - 7t + 10 = 0[/tex]
↓ factoring the quadratic expression
[tex](t - 5)(t - 2) = 0[/tex]
↓ splitting into two equations
[tex]t-5=0[/tex] or [tex]t-2=0[/tex]
[tex]t = 5[/tex] or [tex]t = 2[/tex]
So, the ball will be 160 feet from the ground at 2 seconds and 5 seconds after being thrown.
Further Note
It makes sense that there are two solutions because the position function is a quadratic, meaning it takes the shape of a parabola. Since y = 160 is smaller than the y-coordinate of the vertex, it appears twice on the curve.
