SOLUTION: The velocity is given as {{{V=3t^2-39t+108}}}. Find the times when the velocity is positive

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Question 138680: The velocity is given as V=3t%5E2-39t%2B108. Find the times when the velocity is positive
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
V=3t%5E2-39t%2B108 Start with the given equation


0=3t%5E2-39t%2B108 Set the equation equal to zero

3%28t-9%29%28t-4%29=0 Factor the left side


Now set each factor equal to zero:
t-9=0 or t-4=0

t=9 or t=4 Now solve for t in each case


So the critical values are:

t=9 or t=4



3t%5E2-39t%2B108%3E0 Set the expression greater than zero (remember we want a positive velocity)





Now set up a number line and plot the critical values on the number line

number_line%28+600%2C+-10%2C+10%2C9%2C4%29



So let's pick some test points that are near the critical values and evaluate them.


Let's pick a test value that is less than 4 (notice how it's to the left of the leftmost endpoint):

So let's pick t=3

3t%5E2-39t%2B108%3E0 Start with the given inequality


3%283%29%5E2-39%283%29%2B108%3E+0 Plug in t=3


18%3E+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.

So part our solution in interval notation is ()


However, since a negative time is not possible, the interval is really [)





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Let's pick a test value that is in between 4 and 9:

So let's pick t=6

3t%5E2-39t%2B108%3E0 Start with the given inequality


3%286%29%5E2-39%286%29%2B108%3E+0 Plug in t=6


-18%3E+0 Evaluate and simplify the left side

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.


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Let's pick a test value that is greater than 9 (notice how it's to the right of the rightmost endpoint):

So let's pick t=10

3t%5E2-39t%2B108%3E0 Start with the given inequality


3%2810%29%5E2-39%2810%29%2B108%3E+0 Plug in t=10


18%3E+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.
So part our solution in interval notation is ()





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Summary:

So the solution in interval notation is:


[) ()



So when the time is in between 0 to 4 seconds, or after 9 seconds, then the velocity is positive.