Question

WILL GIVE MEDAL!!! I NEED ANSWER ASAP!!! Lukalu is rappelling off a cliff. The parametric equations that describe her horizontal and vertical position as a function of time are x(t)=8t and y(t)=-16^2+100. . How long does it take her to reach the ground? How far away from the cliff is she when she lands? Remember to show all of the steps that you use to solve the problem.

Answer 1

The horizontal function is x(t) = 8·t
The vertical function is y(t) = -16·t² + 100

A) The answer is: it takes her 2.62 time units to reach the ground.

The time taken to move horizontally is equal to the time taken to move vertically from the cliff to the ground, therefore, we can set:
8·t = -16·t² + 100

and solve for t:
16·t² + 8·t - 100 = 0
4(4·t² + 2·t - 25) = 0
4·t² + 2·t - 25 = 0

$t_{1,2} = \frac{- \frac{b}{2} \pm \sqrt{(\frac{b}{2})^{2} - ac } }{a}$
$t_{1,2} = \frac{- \frac{2}{2} \pm \sqrt{(\frac{2}{2})^{2} - (4)(-25) } }{4}$

t₁ = (-1 + √101) / 4 = 2.62 
t₂ = (-1 - √101) / 4 = - 2.76

We cannot accept negative times, therefore the solution is t = 2.62 time-units.

B) The answer is: she lands 20.96 distance-units from the cliff.

In order to find the distance travelled, substitute the time taken found in point A into the equation of the horizontal movement (using the vertical one you would calculate the height of the cliff):
x(t) = 8·t
x(2.62) = 8 · 2.62 = 20.96

Hence, Lukalu lands 20.96 distance-units from the cliff.