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

Question 138680: The velocity is given as . Find the times when the velocity is positive
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation

Set the equation equal to zero

Factor the left side

Now set each factor equal to zero:
or

or Now solve for t in each case

So the critical values are:

or

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

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 (notice how it's to the left of the leftmost endpoint):

So let's pick

Start with the given inequality

Plug in

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 [

)

---------------------------------------------------------------------------------------------

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

So let's pick

Start with the given inequality

Plug in

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.

---------------------------------------------------------------------------------------------

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

So let's pick

Start with the given inequality

Plug in

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 (

)

---------------------------------------------------------------------------------------------

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.
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