Respuesta :
Answer:
a. 30 feet
b. 2.368 s
Explanation:
a)
The initial position of the grapefruit is 5 feet above the ground when it is released. Its height at time t is given by
y = h(t) = -16t^2 + 40t + 5.
If we plot this function on a graph, we would get a parabola. The vertex of the parabola represents the maximum height reached by the grapefruit before reaching the ground. We would calculate the x coordinate of the vertex of the parabola by applying the formula,
x = - b/2a
Recall, the standard form of a quadratic equation,
y = ax^2 + bx + c
By comparing both equations,
a = - 16
b = 40
c = 5
Thus,
x = - 40/2* - 16 = - 40/- 32 = 1.25
We would find the y coordinate of the vertex by substituting t = 1.25 into the original equation. We have
h(1.25) = -16(1.25)^2 + 40(1.25) + 5
h(1.25) = - 25 + 50 + 5
h(1.25) = 30
The height is 30 feet
b) When the grapefruit reaches the ground, h = 0
Substituting h = 0 into the original equation, we have
-16t^2 + 40t + 5 = 0
We would apply the quadratic formula which is expressed as
[tex]\begin{gathered} x\text{ = }\frac{-\text{ b }\pm\sqrt{b^2-4ac}}{2a} \\ x\text{ = }\frac{-\text{ 40 }\pm\sqrt{40^2-4(-\text{ 16}\times5)}}{2\times-16} \\ x\text{ = }\frac{-\text{ 40 }\pm\sqrt{1280}}{-\text{ 32}} \\ x\text{ = }\frac{-\text{ 40 + 35.777}}{-\text{ 32}}\text{ or x = }\frac{-\text{ 40 - 35.777}}{-\text{ 32}} \end{gathered}[/tex]t = 0.132 or t = 2.368
Since it takes 1.25s to reach maximum height, the reasonable time for it to reach the ground is 2.368 s
