Solve each problem. See Example 7. Velocity of an Object The velo...

Question

Solve each problem. See Example 7. Velocity of an Object The velocity of an object, v, after t seconds is given by v=3t^2-18t+24.Find the interval where the velocity is negative.

Accepted Answer

Hey everyone here we are asked to estimate the time when the velocity of the car is negative. When the car is moving with a velocity V equal to 40 squared plus 12 t minus 16. So to begin solving this problem, we first need to write our given equation here are given equation of velocity as an inequality and we can do this by recalling that. We're finding the time when velocity is negative. So that means that is when V is less than zero. So we can take this equation or expression for V and set it less than zero. So our inequality is 40 squared plus 12, T minus 16 is less than zero. So now we want to find our values 40. So we actually need to take the expression on the left hand side of the inequality and set it equal to zero. So we have four T squared plus 12 T minus 16 is equal to zero. So now we can solve for t by factoring and we can first factor out a four from each of our terms. So we have four times the quantity of t squared plus three t minus four is equal to zero. And now we can further factor the expression within the parentheses. So we have four times the quantity of T -1 times the quantity of T plus four is equal to zero. And now to find our values 40 we can set each expression in each set of parentheses equal to zero. So we can start with t minus one and again we need to set it equal to zero and now solving 40 we need to add one to both sides and we have T is equal to one. Now for our other part of our expression we have T plus four is equal to zero, solving for T. We need to subtract four from both sides and we get T is equal to negative four. Now with the solutions for T. That we found, we can write our we can write time in interval notation, so we can begin by going from negative infinity. And the next solution that is closest to negative infinity is negative four. So we go from negative infinity to negative four and now we know that infinity is always excluded. So it gets closed by a parenthesis also negative four gets a parentheses to close it because we need the velocity to be less than zero and negative four makes velocity equal to zero And now our next interval we can go from negative 4-1. And again both of these values get closed by parentheses since they are excluded and next we can go from one all the way to positive infinity. And again both of these terms are closed by parentheses. So we have found each of the intervals that can express time in this scenario but now we need to find which of these intervals can give us a that can give us a negative velocity. So to do this we need to set up a table where we take our intervals, we take our intervals and then we input a test value, We choose a test value and input it into our expression into our velocity equation of 40 squared plus 12 T -16. And from there we get a conclusion where we determine the sign of the velocity. So we can begin with our first interval where we go from negative infinity to negative four. We can input the test value of negative five and now we input negative five into our t variables in our formula here. So we have four times the quantity of negative five squared plus 12 Times the quantity of -5 -16. And we see after simplifying that this expression is equal to 24 and we know that 24 is greater than zero. So we will not be using this interval because again we are looking for when velocity is negative and 24 is greater than zero. So this interval is not with in our solution next we can test the next interval where we go from negative 4 to and we can put in the test value of zero. So we have four times zero squared plus times zero minus 16. Now simplifying we get negative 16 which we know is less than zero. So this interval will be included in our solution. Now we need to test our last interval here where we have, we go from one to positive infinity We can choose a test value of two. And now in putting two into our equation from earlier we have four times the quantity of two squared plus 12 times 2 -16. Simplifying. We see that this is equal to 24 which again Is greater than zero. So this interval will not be included in our solution. So now looking at our interval that works again, we have the interval negative 4-1. Well we know that time cannot be negative. So we will actually be going from 0 to 1. And with this we have found our time interval that gives us a negative velocity. So we are left with our final answer here. A time interval from 0 to 1 which is answer choice B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each problem. See Example 7. Velocity of an Object The velocity of an object, v, after t seconds is given by v=3t^2-18t+24.Find the interval where the velocity is negative.

Key Concepts

Quadratic Functions

Finding Roots

Interval Notation

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