SOLUTION: Jacob is standing at the top of an 80 ft-high cliff and throws a rock upwards at 64 ft/sec Write a model that describes the trajectory of the rock. How many seconds until the

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Question 1150174: Jacob is standing at the top of an 80 ft-high cliff and throws a rock upwards at 64 ft/sec
Write a model that describes the trajectory of the rock.
How many seconds until the max height is reached?
What will be the maximum height?
How many seconds until the rock hits the ground?
How high is the rock after 3 seconds?
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Write a model that describes the trajectory of the rock. 
h%28t%29=h%280%29%2Bv%280%29t-16t%5E2

h%28t%29=80%2B64t-16t%5E2
How many seconds until the max height is reached?
 

The easiest way is to use the vertex formula, for the maximum height 
is reached at the vertex which has t coordinate.  We put the 
quadratic in standard form at²+bt+c

h%28t%29=-16t%5E2%2B64t%2B80

-b%2F%282a%29=%28-%2864%29%29%2F%282%28-16%29%29=%28-64%29%2F%28-32%29+=+2
0=64t-16t%5E2
0=16t%284-t%29
16t=0  4-t=0
t=0    4=t

It take 4 seconds for the rock to get back even with Jacob, so it
reached its maximum height in half that time or 2 seconds.  But
the vertex formula is the easiest.
What will be the maximum height?
 

The rock's maximum height is reached when t=2 seconds.

h%28t%29=80%2B64t-16t%5E2
h%282%29=80%2B64%282%29-16%282%29%5E2
h%282%29=80%2B128-16%284%29
h%282%29=80%2B128-64
h%282%29=144ft
How many seconds until the rock hits the ground?

That's when h(t) = 0

We set h(t)=0 and solve for t

h%28t%29=80%2B64t-16t%5E2
0=80%2B64t-16t%5E2

Divide through by 16

0=5%2B4t-t%5E2
0=%285-t%29%281%2Bt%29
0=5-t    0=1%2Bt
t=5      -1=t

Ignore the negative answer.

5 seconds.
How high is the rock after 3 seconds?

This is h(3)

h%28t%29=80%2B64t-16t%5E2
h%283%29=80%2B64%283%29-16%283%29%5E2
h%283%29=80%2B192-16%289%29
h%283%29=80%2B192-144
h%283%29=128ft

Edwin

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

In addition to the remarkable solution by Edwin,  see also the lessons
    - Problem on a projectile moving vertically up and down
    - Problem on an arrow shot vertically upward
    - Problem on a ball thrown vertically up from the top of a tower
    - Problem on a toy rocket launched vertically up from a tall platform
in this site.

Consider these lessons as your textbook,  handbook,  tutorials and  (free of charge)  home teacher.
Read them attentively and learn how to solve this type of problems once and for all.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Projectiles launched/thrown and moving vertically up and dawn".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

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