Question 1192920
.
{{{ h(t) = c - (d-4t)^2 }}}

At time t=0, a ball was thrown upward from an initial height of 4 feet. 
The ball's height, in feet, after t seconds is given by the function h above, 
in which both c and d are positive constants. If the ball reached its maximum height 
of 104 feet at time t=3, what is the height, in feet, of the ball at time t=1.5?
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<pre>
In this problem, the standard quadratic function of height, h(t) = -16t^2 + ut + h0,  is presented

in vertex form, and your first task is to identify the parameters of this vertex form.



    I will help you to identify these parameters.



First, the term "c" represents the highest position of the ball, which is given in the problem as 104 feet.

So, c = 104 feet.



Second, d - 4t = 0  determines the time "t", when the highest position is reached.

The problem says that the maximum height is reached at t= 3 seconds;  hence, d= 4t = 4*3 = 12 seconds.



    Now you know everything about your function: it is  h(t) = 104 - (12-4t)^2.



Now to answer the problem's question, you simply substitute t= 1.5 seconds in the last formula.

You get then


    h(1.5) = 104 - (12 - 4*1.5)^2 = 68 feet.    <U>ANSWER</U>


Thus, the solution of this tangled problem is completed.
</pre>

Solved, explained and completed.


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In this site, &nbsp;there is a bunch of lessons on a projectile thrown/shot/launched vertically up


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Introductory-lesson-on-a-projectile-thrown-shot-launched-vertically-up.lesson>Introductory lesson on a projectile thrown-shot-launched vertically up</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-projectile-moving-vertically.lesson>Problem on a projectile moving vertically up and down</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-projectile-shooted-vertically-upward.lesson>Problem on an arrow shot vertically upward</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Typical-problems-on-an-projectile-moving-vertically-up-and-down.lesson>Problem on a ball thrown vertically up from the top of a tower</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-toy-rocket-launched-vertically-up--from-the-top-of-a-platform.lesson>Problem on a toy rocket launched vertically up from a tall platform</A>


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


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


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



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


to your archive and use it when it is needed.



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<H3>Post-solution notes at the end:</H3>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(a)  &nbsp;&nbsp;the initial condition &nbsp;"at &nbsp;time t=0, &nbsp;a ball was thrown upward from an initial height of &nbsp;4 &nbsp;feet" 

  
     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;is not used in the solution. &nbsp;So, &nbsp;this condition is &nbsp;EXCESSIVE &nbsp;and &nbsp;UNNECESSARY.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;By the way, &nbsp;this condition is &nbsp;INCONSISTENT &nbsp;with the rest of the problem, 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;so &nbsp;IT &nbsp;MUST &nbsp;BE &nbsp;EXCLUDED &nbsp;from the problem, &nbsp;for clarity.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(b)  &nbsp;&nbsp;To solve the problem, &nbsp;it is assumed that the reader/(the student) firmly knows the prerequisites

     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;that are given in the lessons listed in my post.


     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This knowledge is a &nbsp;NECESSARY &nbsp;condition to solve the problem.


     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If you know these prerequisites, &nbsp;you are able to solve the problem and to understand the solution.


     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If you do not know them, &nbsp;you will find yourselves in difficult position.