Solve each problem. See Example 7. A baseball is hit so that its ...

Question

Solve each problem. See Example 7. A baseball is hit so that its height, s, in feet after t seconds is s=-16t^2+44t+4. For what time period is the ball at least 32 ft above the ground?

Accepted Answer

Hey, everyone here we are given the height of the leather ball after time T and it is given as H is equal to negative eight T squared plus 22 T plus two. And whereas to estimate the time where the leather ball is at least 14 ft above the ground, so this last piece of information where the ball is the a minimal Of 14 ft above the ground. This tells us that age again is a minimum of 14 ft. So it must be greater than or equal to 14 ft. And so now we can write in inequality to describe this situation using this last piece of given information. So all we need to do is substitute in this expression for h in our inequality and we have negative eight T squared plus 22 T plus two is greater than or equal to 14. And now we just need to solve this inequality. So we want to begin by getting the coefficient of T squared to be positive. And we do this by multiplying both sides of our inequality by negative one. But since we are multiplying by a negative value, we need to make sure we flip our sign. So our inequality becomes a T squared minus 22 T minus two is less than or equal to negative 14. And now we just want to get the right hand side of our, our inequality to be zero. So we just need to add 14 to both sides. And this gives us the inequality of 80 squared minus 22 T plus 12 is less than or equal to zero. And now to find values 40 we need to set the left hand side equal to zero. So we have a T squared minus 22 T plus 12 is equal to zero. So now we just need to find values 40 by factoring the left hand side. So we can begin by first factoring out a common factor from each term, which in this case is going to be two. So we have two times the quantity of four T squared minus 11 T plus six is equal to zero. And now we just need to factor the expression inside the parentheses further. So we have two times the quantity of four T -3 times the quantity of T -2 is equal to zero. And now we just need to find the values of T that set the left hand side expression equal to zero. So that means we just need to take each expression in each set of parentheses and individually set them equal to zero. So we can start with our first set of parentheses where we have four T minus three. And again, setting this equal to zero, we now have to solve for T. So first, we need to add three to both sides. So we, we have four, T is equal to three. And now we just need to get T alone by dividing both sides by four. And we see that T is equal to 3/4 which is the decimal equivalent of 0.75. And now we can do the same thing with the second part of our expression where we have T minus two. And again, we set this equal to zero and solve for T. So we need to add two to both Sides. And we see that T is also equal to two. So now we see that when T is equal to two or at two seconds in our time interval here, we will get a height of zero fee and the same thing applies for when time is at 0. seconds. So now we just want to take these two solutions for tea and express time using intervals. So we can begin by going from negative infinity to our closest next solution here, which will be 0.75. Well, first we know that negative infinity gets closed by a parenthesis because infinity is always excluded. But now our 0.75 gets closed by a bracket. Because if we recall our original inequality symbol, we have a non strict inequality. So our 0.75 is included. So now we go to the next interval, we go from 0.75 to our next solution, which is two. So we go from 0.75 to two. And now both of these values for the same reason are in included and so closed by brackets. And lastly, we go from two to positive infinity and two again gets closed by a bracket, but infinity is closed by a parenthesis. So now we just need to find the sign of the expression eight T squared minus 22 T Plus 12 using each of these intervals. So we just need to set up a table where we have our interval, we have each of our intervals from before and we choose a test value. And then we plugged that test value into our equation here and determine our conclusion by assessing the sign. So doing this, we can start with our first interval that we found, which is from negative infinity to 0.75. And now we choose a test value that's within this interval. So we can choose the test value zero. And now we take zero and plug it into T for our equation above. And we see that we have eight times zero squared minus 22 times zero plus 12. And we see that this is equal to 12, which is greater than zero. So We see that since we got a solution greater than zero, that this interval does not fit our solution. Because as we recall from earlier, we are looking for a value that's less than or equal to zero. So now we can move on to our next interval where we went from 0.75-2. And we can plug in a test value that's within this interval, we can choose the value one. Now plugging one into our equation, we have eight times one squared minus times one plus 12. And now simplifying we get negative two, which is less than zero and less than zero does fit within our inequality. So, so this interval does work for our solution. And now we just want to test our last interval here where we went from two to positive infinity. And now we can choose a test value here of let's say three. And now plugging three into our equation, we have eight times three squared minus times three plus 12. And now simplifying we get 18 which is also greater than zero. And as we had with our first interval, we know that greater than zero does not fit within our solution. So this last interval is not included in our solution. So now looking at our three intervals, we see that we only have one interval that works for our given situation. And therefore we know that the required time interval for the ball to be at least 14 ft above the ground is from 0.75 seconds to two seconds. So with this, we have found or estimated the time when the leather ball is at least ft above the ground. And therefore we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each problem. See Example 7. A baseball is hit so that its height, s, in feet after t seconds is s=-16t^2+44t+4. For what time period is the ball at least 32 ft above the ground?

Key Concepts

Quadratic Functions

Vertex of a Parabola

Inequalities

Watch next