Question

A rocket is fired upward from some initial distance above the ground. Its height in​ feet, h,above the​ ground, t seconds after it is​ fired, is given by h=−16t^2+96t+2560.

What is the rocket's maximum height?

How long does it take for the rocket to reach its max height?

After it's fired, the rocket reaches the ground at t=__ sec.

Answer 1

check the picture below.

$\textit{vertex of a vertical parabola, using coefficients} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{-16}x^2\stackrel{\stackrel{b}{\downarrow }}{+96}x\stackrel{\stackrel{c}{\downarrow }}{+2560} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right) \\\\\\ \left(-\cfrac{ 96}{2(-16)}~~~~ ,~~~~ 2560-\cfrac{ (96)^2}{4(-16)}\right)\implies \left(-\cfrac{96}{-32} ~~,~~2560-\cfrac{9216}{-64} \right)$

$\left( 3~~,~~2560+144 \right)\implies \underset{maximum~height}{\stackrel{\textit{it took this long}}{(\stackrel{\downarrow }{3}~~,~~\underset{\uparrow }{2704})}} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf h(t)=-16t^2+96t+2560\implies 0=-16t^2+96t+2560 \\\\\\ 0=-16(t^2-6t-160)\implies 0=t^2-6t-160 \\\\\\ 0=(t-16)(t+10)\implies t= \begin{cases} \boxed{16}\\ -10 \end{cases}$

"t" is an amount in seconds, for this specific phenomena, it cannot be negative, so -10 is not a feasible value.

Answer 2

The rocket's maximum height is 2704 feet, it takes 3 seconds for the rocket to reach its max height, and the rocket reaches the ground at t= 16 sec

What is a parabola?

It is defined as the graph of a quadratic function that has something bowl-shaped.

We have a relation between height and time as follows:

$\rm h = -16t^2+96t+2560$

The above equation represents a parabola equation thus its vertex:

$\rm (\frac{-b}{2a} , c - \frac{b^2}{2a} )$

Here a = -16, b = 96, and c = 2560

So the maximum height at:

= -96/(2×(-16)

= 96/32

= 3 seconds

And maximum height after 3 seconds:

$\rm h = -16\times3^2+96\times3+2560$

h = 2704 feet

At y = 0 the rocket reaches the ground putting this value in the parabolic equation:

$\rm 0 = -16t^2+96t+2560$

After solving we will get:

t = 16 or t = -10

Time cannot be a negative quantity so taking t = 16 seconds

Thus, the rocket's maximum height is 2704 feet, it takes 3 seconds for the rocket to reach its max height, and the rocket reaches the ground at t= 16 sec

Know more about the quadratic equation here:

brainly.com/question/2263981

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