Question

a. write a function f that models the height of the snowboarder over time.

f(x)=

how long is the snowboarder in the air when he lands 7 feet below the base of the jump?

seconds

Explanation:

Step1: Assume a quadratic - function form

The height - time relationship of an object in vertical motion under the influence of gravity can be modeled by a quadratic function of the form $f(x)=ax^{2}+bx + c$. Without further information about the initial velocity and initial height, if we assume the initial height is $h_0$ and the initial vertical velocity is $v_0$, and the acceleration due to gravity $g = - 32$ ft/s² (in English units), the general formula for the height $h$ of an object as a function of time $t$ is $h(t)=-\frac{1}{2}gt^{2}+v_0t + h_0$. If we assume the snow - boarder starts at height $h_0 = 0$ and has an initial vertical velocity $v_0$, then $f(x)=-\ 16x^{2}+v_0x$. Let's assume the snow - boarder starts from rest vertically ($v_0 = 0$) for simplicity, so $f(x)=-16x^{2}$.

Step2: Solve for the time when the height is $-7$

Set $f(x)=-7$. So, $-16x^{2}=-7$.
Then $x^{2}=\frac{7}{16}$.
Taking the square root of both sides, we get $x=\pm\sqrt{\frac{7}{16}}$. Since time $x\geq0$, $x = \frac{\sqrt{7}}{4}\approx\frac{2.646}{4}=0.6615$.

Answer:

$f(x)=-16x^{2}$
$0.66$ (rounded to two decimal places)