pre-calculus, vertex/matrixes

Question

The path of a piece of candy as it flies from a child’s hand to their mouth can be described using the equation h(t) = 3t − 5 − 0.15t^2 , where t represents the time in milliseconds since the candy was thrown, and h(t) represents the height of the candy in inches.

1. Can the vertex of this be shown?
2. If h(t) = 0 represents the height of the child’s mouth, determine how long it will take for them to catch the candy.

Answer 1

1)

h(t) is an upside down parabola with the vertex at t = -b/(2a) = -3/(-.3) = 10.

h(10) = 10, so the vertex is at (10,10)

2)

h(0) = -5

for 3t -5 -0.15t² = -5, t(3-.15t) = 0 or t = 0, 20

So it will take 20 ms for the candy to reach the child's mouth

Answer 2

vertex= 3/0.3=10

-0.15t^2+3t-5=0

t^2-20t+100/3=0

t^2-20t+100=200/3

(t-10)^2=200/3

t-10=(200/3)^0.5 or -(200/3)^0.5

t=18.16. or -1.83

t should be positive, so t=18.16