Respuesta :
Answer:
Time taken by the ball to reach its maximum height is 3.4 sec
maximum height reached by the ball is 185.6ft
Step-by-step explanation:
h(t) = -16[tex]t^{2}[/tex] + 109t
height is maximum ⇔ [tex]\frac{d}{dt}[/tex](h) = 0 and [tex]\frac{d^{2} }{dt^{2} }[/tex](h) < 0
[tex]\frac{d}{dt}[/tex](h) = 0 ⇒ -32t + 109 = 0
⇒t = [tex]\frac{109}{32}[/tex] ⇒ t = 3.4sec
[tex]\frac{d^{2} }{dt^{2} }[/tex](h) = -32 < 0
∴h(t) is maximum at t = 3.4
and maximum height is h(3.4) = -16×[tex]3.4^{2}[/tex] + 109×3.4 ⇒ h = 185.6ft
Answer:
The maximum height with rounding by the nearest tenth is [tex]280.56\ m[/tex]
Explanation:
Initial velocity of ball [tex]=109\ ft/sec[/tex]
Quadratic function [tex]$h(t)=-16t x^{2} +109t$[/tex]
Maximum height[tex]=?[/tex]
Step 1:
This ball has its height related to time by a parabola with the equation:
[tex]$s=\frac{1}{2} \times a \times t^{2}+v_{0} \times t+s_{0}$[/tex]
where [tex]$s$[/tex] is the height so is the starting height, [tex]v_{0}[/tex] is its starting velocity [tex]$=109 \ {ft} / \mathrm{s}$[/tex]
[tex]$a$[/tex] is the acceleration of gravity, which is [tex]$-16 \ {ft} / \mathrm{s}$[/tex] every second.
Step 2:
Time required to reach its maximum height:
[tex]$v=a^{*} t+v_{0}[/tex] (its velocity is the acceleration of gravity × time [tex]+ v_0[/tex]) [tex]$v$[/tex] will slow down from [tex]$109 \ {ft} / \ s[/tex] to [tex]$0 \mathrm{ft} / \mathrm{s}$[/tex]
And it slowing down at [tex]$16 \mathrm{ft} / \mathrm{s}$[/tex] :
[tex]t(s)=0,1,2,3,4,5,6[/tex]
[tex]V (m/s)=109,93,77,61,45,29,13[/tex]
Before [tex]$7 \mathrm{~s}$[/tex], it will hit the ground.
Exact calculation [tex]$t=\frac{109 f t / s}{16 f t / s}=6.81 s$[/tex]
Thus we can say at the height of,
[tex]$\mathrm{t}_{\max }=\frac{6.81}{2} \\=3.4\ sec[/tex]
Step 3:
Plug the [tex]t_m_a_x[/tex] sec value back into equation:
[tex]$h(t)=t \cdot(-16 t+109)$[/tex]
[tex]$h_{\max }=3.4 \cdot(-16.4 .1875+109)$[/tex]
[tex]$h_{\max }= 185.6\ m[/tex]
Hence, the maximum height is [tex]185\ m[/tex]
To learn more about vertically upward ball, refer:
- https://brainly.com/question/17925478
- https://brainly.com/question/14614888
