<PRE><B>Question 702202</B>
&lt;br&gt;&lt;font face=&quot;Tahoma&quot;&gt;This equation represents a parabola,&lt;br&gt;

where the vertex is at a maximum,&lt;br&gt;

and it opens down because of the negative coefficient of t.&lt;br&gt;

{{{h=144t-16t^2}}}&lt;br&gt;

{{{h=-16t^2+144t}}} where a=-16, b=144, and c=0&lt;br&gt;

We can easily find the vertex (h,k) if we know the proper formulae:&lt;br&gt;

{{{h=-b/2a}}}&lt;br&gt;

{{{h=-144/(2*-16)}}}&lt;br&gt;

{{{h=9/2}}}&lt;br&gt;

{{{k=f(h)}}}&lt;br&gt;

{{{k=f(9/2)}}}&lt;br&gt;

{{{k=-16(9/2)^2+144*(9/2)}}}&lt;br&gt;

{{{k=324}}}&lt;br&gt;

So our vertex is the point (9/2,324)&lt;br&gt;

We can also find the t-intercepts,&lt;br&gt;

by setting h=0 and solving for t.&lt;br&gt;

In this case, our t-intercepts are 0 and 9.&lt;br&gt;

{{{graph( 600, 600, -1, 10, -10, 350, -16x^2+144x )}}}&lt;br&gt;

By inspection of the graph,&lt;br&gt;

it looks like the period of time that the arrow is over 224 feet&lt;br&gt; 

is between approximately 2 seconds and 7 seconds.&lt;br&gt;

This is about 5 total seconds.&lt;br&gt;

Of course this is only an approximation!  Our graphs are never exact.&lt;br&gt;

To find the exact answer,&lt;br&gt;

we would need to find the points where h equals 224 in our equation,&lt;br&gt;

and then we could find the exact times.&lt;br&gt;

We would do this by setting the equation equal to 224, and solving for t.&lt;br&gt;

This would give us the extremes of the range of t-values where h was greater than 224.&lt;br&gt;

PS It does indeed look like 2 and 7 are the exact t-values we need.&lt;br&gt;

I hope this helps!  Keep practicing!  :)&lt;br&gt;

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