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During a game's intermission, a team mascot launches tightly rolled tee shirts into the stands. The height $h(t),$ in feet, of an airborne tee shirt $t$ seconds after being launched can be approximated by$$h(t)=-15 t^{2}+75 t+10$$After peaking, a rolled-up tee shirt is caught by a fan 70 ft above ground level. For how long was the tee shirt in the air?CAN'T COPY THE GRAPH

$4 \sec$

Algebra

Chapter 6

Polynomial Factorizations and Equations

Section 7

Applications of Polynomial Equations

Graphs and Statistics

Exponents and Polynomials

Equations and Inequalities

Polynomials

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Harvey Mudd College

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this equation represents the height of an airborne T shirt and respect to the time of how long it's been airborne for, and were given that the high is 70 feet and we want to solve for how long? Ah, was the shirt airborne after a reach, its peak value and that key were like he phrased after reaching speed values very important, and we'll get back to it in the end. So let's replace uh, H with 70 feet. So we have 70 is equal to negative 15 t squared plus 75 t plus 10 subtracting 70 on both sides. We yield zero is equal to negative 15 t squared plus 75 T minus 60 now recon most by both sides by a negative one just to get this coefficient in front into a positive quantity, since factoring, it's much easier when you have a polynomial with the highest power as a positive quantity. Now we get zero is equal to 15 t squared minus 75 T plus 60. Now we could find the greatest common factor. So the greatest coming factor between thes three terms is 15 so we have zero Ziegel to 15 times t squared minus five T plus four Now dividing 15 on both sides we get zero is equal to t squared minus five t plus four. And now it's just a simple polynomial that we have to factor. And we could do that using the X method in terms of tea. Now, So we have a time see which is negative, which is one times four and be down here with negative five. So what times? What is equal to four and what, plus what is equal to negative by. So we see that here that is going to be negative four and negative one gives you positive, positive or winning more crime and negative board, plus negative one will give you negative five. So now we have X minus four and X minus one equals zero. To have ecstasy with you Poor and X is equal to one. So we had two different times, but I think she's which was the right time. Excuse me. This is actually E. That's what used to you. So now we had to pick which one is the right time. And to do so, we have to go back to that key phrase that I told you about after peaking so it would look at this diagram and we put it that find here. Say this is some arbitrary height will call this height 20 feet. Initially, we could have 20 feet here than 20 feet here. These are both 20 feet, but at different times, one is before the peak value the big. Now you being here, this maximum peak and the other one being after the peak value, which is this one here after peak and were given that we want to find the time after it reached its peak value. And so, given that information, we can now exclude this one because we don't want to find out when we want to find after at Astro speed value, which is that one there. So in our case, it's going to be the longer time. So we exclude times, you go to one. And now our answer is t

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