1. Jason jumped off a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function h(t) = -16t² + 16t + 480, where t is the time in seconds and h is the height in feet.a. How long did it take for Jason to reach his maximum height?b. What was the highest point that Jason reached?c. Jason hit the water after how many seconds?
Question
- Jason jumped off a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function h(t) = -16t² + 16t + 480, where t is the time in seconds and h is the height in feet.a. How long did it take for Jason to reach his maximum height?b. What was the highest point that Jason reached?c. Jason hit the water after how many seconds?
Solution
a. Para encontrar el tiempo que le tomó a Jason alcanzar su altura máxima, necesitamos encontrar el vértice de la parábola dada por la función h(t) = -16t² + 16t + 480. La fórmula para el tiempo en el vértice de una parábola de la forma ax² + bx + c es t = -b/(2a).
En este caso, a = -16 y b = 16. Entonces,
t = -16 / (2 * -16) t = -16 / -32 t = 0.5
Por lo tanto, le tomó 0.5 segundos a Jason alcanzar su altura máxima.
b. Para encontrar la altura máxima, evaluamos la función h(t) en t = 0.5.
h(0.5) = -16(0.5)² + 16(0.5) + 480 h(0.5) = -16(0.25) + 8 + 480 h(0.5) = -4 + 8 + 480 h(0.5) = 484
La altura máxima que Jason alcanzó fue de 484 pies.
c. Para encontrar el tiempo en que Jason golpeó el agua, necesitamos encontrar los valores de t cuando h(t) = 0.
-16t² + 16t + 480 = 0
Podemos resolver esta ecuación cuadrática usando la fórmula cuadrática, t = [-b ± √(b² - 4ac)] / 2a, donde a = -16, b = 16, y c = 480.
t = [-16 ± √(16² - 4(-16)(480))] / 2(-16) t = [-16 ± √(256 + 30720)] / -32 t = [-16 ± √30976] / -32 t = [-16 ± 176] / -32
Esto nos da dos soluciones:
t₁ = (-16 + 176) / -32 t₁ = 160 / -32 t₁ = -5 (no tiene sentido en este contexto)
t₂ = (-16 - 176) / -32 t₂ = -192 / -32 t₂ = 6
Por lo tanto, Jason golpeó el agua después de 6 segundos.
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