document.write( "Question 138680: The velocity is given as $\"V=3t%5E2-39t%2B108\"$ . Find the times when the velocity is positive \n" ); document.write( "

Algebra.Com's Answer #101193 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"V=3t%5E2-39t%2B108\"$ Start with the given equation\r
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\n" ); document.write( "\n" ); document.write( " $\"0=3t%5E2-39t%2B108\"$ Set the equation equal to zero\r
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\n" ); document.write( "\n" ); document.write( " $\"3%28t-9%29%28t-4%29=0\"$ Factor the left side \r
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\n" ); document.write( "\n" ); document.write( "Now set each factor equal to zero:\r
\n" ); document.write( "\n" ); document.write( " $\"t-9=0\"$ or $\"t-4=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"t=9\"$ or $\"t=4\"$ Now solve for t in each case\r
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\n" ); document.write( "\n" ); document.write( "So the critical values are:
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\n" ); document.write( "\n" ); document.write( " $\"t=9\"$ or $\"t=4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"3t%5E2-39t%2B108%3E0\"$ Set the expression greater than zero (remember we want a positive velocity)\r
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\n" ); document.write( "\n" ); document.write( "Now set up a number line and plot the critical values on the number line\r
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\n" ); document.write( "\n" ); document.write( " $\"number_line%28+600%2C+-10%2C+10%2C9%2C4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So let's pick some test points that are near the critical values and evaluate them.\r
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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is less than $\"4\"$ (notice how it's to the left of the leftmost endpoint):\r
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\n" ); document.write( "\n" ); document.write( "So let's pick $\"t=3\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"3t%5E2-39t%2B108%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"3%283%29%5E2-39%283%29%2B108%3E+0\"$ Plug in $\"t=3\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"18%3E+0\"$ Evaluate and simplify the left side\r
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\n" ); document.write( "\n" ); document.write( "Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.\r
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\n" ); document.write( "\n" ); document.write( " So part our solution in interval notation is (

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\n" ); document.write( "\n" ); document.write( "However, since a negative time is not possible, the interval is really [

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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is in between $\"4\"$ and $\"9\"$ :\r
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\n" ); document.write( "\n" ); document.write( "So let's pick $\"t=6\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"3t%5E2-39t%2B108%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"3%286%29%5E2-39%286%29%2B108%3E+0\"$ Plug in $\"t=6\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"-18%3E+0\"$ Evaluate and simplify the left side\r
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\n" ); document.write( "\n" ); document.write( "Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.\r
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\n" ); document.write( "\n" ); document.write( "Let's pick a test value that is greater than $\"9\"$ (notice how it's to the right of the rightmost endpoint):\r
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\n" ); document.write( "\n" ); document.write( "So let's pick $\"t=10\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"3t%5E2-39t%2B108%3E0\"$ Start with the given inequality\r
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\n" ); document.write( "\n" ); document.write( " $\"3%2810%29%5E2-39%2810%29%2B108%3E+0\"$ Plug in $\"t=10\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"18%3E+0\"$ Evaluate and simplify the left side\r
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\n" ); document.write( "\n" ); document.write( "Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.\r
\n" ); document.write( "\n" ); document.write( " So part our solution in interval notation is (

)

(

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\n" ); document.write( "\n" ); document.write( "So when the time is in between 0 to 4 seconds, or after 9 seconds, then the velocity is positive. \n" ); document.write( "

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