Quadratic Functions and Projectile Motion Models

Question

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation H(t)=−16t^2+96t+112.

How long will it take, in seconds, for the rock to hit the ocean? 

Answer 1

It takes 7 seconds for the rock to hit the ocean.

The ocean is taken as height 0 and upward as the positive direction. The -16 represents the acceleration of gravity and is negative because it is downward. 96 is the initial velocity and 112 is the starting height.

0 = -16t^2 +96t + 112

taking out the common factor -16

-16(t^2 - 6t - 7) = 0

factoring the trinomial

-16(t - 7)(t + 1) = 0

We take t = 7 sec as the answer and reject t = -1 as an impossible answer because it would represent negative time.

Answer 2

The equation of the rock's height above the ocean is given as H(t) = -16 t^2 +96t +112

To determine how long it will take for the rock to hit the ocean, solve for t

H(t) = -16 t^2 +96t +112 = 0

I have simplified this problem by factoring out the common factor of 16 in each term, yielding

H(t) = 16( - t^2 +6 t +7) = 0

using the Quadratic Formula I have two real roots:( - 6 +/- sqrt (( 6^2) -4(-1)(7))) /2(-1)

which simplifies to ( -6 +/ - sqrt( 36 +28)) / (-2)

which simplifies again to ( -6 +/ - sqrt( 64)) / (-2)

which simplifies again to ( -6 +/ - 8)) / (-2)

conclude possible roots are -14/-2 = +7 or +2/-2 = -1.

The negative root makes no sense. So the solution is 7 seconds.