Question

1. the height of a model rocket is launched straight upward off the ground at an initial velocity of 200 feet per second can be modeled by the function, $h(t)=-16t^{2}+200t$. what is the maximum height reached by the rocket? when will the rocket reach its maximum height?

Explanation:

Step1: Identify the coefficients

The height - function is $h(t)=-16t^{2}+200t$, where $a = - 16$, $b = 200$, $c = 0$.

Step2: Find the time $t$ at which the rocket reaches its maximum height

The time $t$ at which a quadratic function $y = ax^{2}+bx + c$ reaches its maximum (when $a<0$) is given by the formula $t=-\frac{b}{2a}$. Substitute $a=-16$ and $b = 200$ into the formula:
$t=-\frac{200}{2\times(-16)}=\frac{200}{32}=6.25$ seconds.

Step3: Find the maximum height

Substitute $t = 6.25$ into the height - function $h(t)=-16t^{2}+200t$:
$h(6.25)=-16\times(6.25)^{2}+200\times6.25$
$=-16\times39.0625 + 1250$
$=-625+1250$
$=625$ feet.

Answer:

The rocket reaches its maximum height of 625 feet at $t = 6.25$ seconds.