Complete Question:
Alejandro jumped from a cliff into the ocean in Acapulco while vacationing with some friends.
[tex]h(t)=-16t^2 + 16t + 480[/tex]
Where t is the time in seconds and h is the height in feet
(a) How long did it take for Alejandro to reach his maximum height
(b) What was the highest point Alejandro reached
(c) Alejandro hits the water after how many seconds
Answer:
(a) 0.5 seconds
(b) 484 feet
(c) 6 seconds
Step-by-step explanation:
Given
[tex]h(t)=-16t^2 + 16t + 480[/tex]
Solving (a): Time to reach maximum height
This is the maximum of the function and it is calculated using:
[tex]t = -\frac{b}{2a}[/tex]
Where
[tex]a = -16; b = 16; c =480[/tex]
So:
[tex]t = -\frac{16}{2*-16}[/tex]
[tex]t = \frac{16}{32}[/tex]
[tex]t = 0.5[/tex]
Solving (b): Highest point reached
Time to reach the highest point is 0.5.
So, the highest point is: h(0.5)
[tex]h(t)=-16t^2 + 16t + 480[/tex]
[tex]h(0.5) = -16 * 0.5^2 + 16 * 0.5 + 480[/tex]
[tex]h(0.5) = 484[/tex]
Solving (c): Time he hits water.
At this point, h(t) = 0
So;
[tex]h(t)=-16t^2 + 16t + 480[/tex]
[tex]-16t^2 + 16t + 480 = 0[/tex]
Factorize
[tex]-16(t^2 - t -30) = 0[/tex]
Divide both sides by -16
[tex]t^2 - t -30 = 0[/tex]
Expand
[tex]t^2 + 5t - 6t - 30 = 0[/tex]
Factorize
[tex]t(t + 5)-6(t + 5) =0[/tex]
[tex](t -6)(t + 5) =0[/tex]
[tex]t =6\ or\ t = -5[/tex]
Time can't be negative.
So:
[tex]t = 6[/tex]
