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After seconds, the height (in feet) of = baseball pitched into the air is given by the functionH()= -16t2+20t+8 What is its height before pitch? Fer €What is ...

Question

After seconds, the height (in feet) of = baseball pitched into the air is given by the functionH()= -16t2+20t+8 What is its height before pitch? Fer €What is the time when this ball reaches its highest point? 0.625 CaeWhat is its highest point? 7(8)+3(3)+8 STor 142s [cWhen does the ball hit the ground? (Round to decima places_thoce fuinctions to answer #18 and 19_

After seconds, the height (in feet) of = baseball pitched into the air is given by the function H()= -16t2+20t+8 What is its height before pitch? Fer € What is the time when this ball reaches its highest point? 0.625 Cae What is its highest point? 7(8)+3(3)+8 STor 142s [c When does the ball hit the ground? (Round to decima places_ thoce fuinctions to answer #18 and 19_



Answers

A baseball is thrown straight up, and its height as a function of time is given by the formula $h=-16 t^{2}+32 t$ (where $h$ is in feet and $t$ is in seconds). (a) Find the height of the ball when $t=1$ sec and when $t=3 / 2 \mathrm{sec}$ (b) Find the maximum height of the ball and the time at which that height is attained. (c) At what times is the height $7 \mathrm{ft} ?$

So here we have a function h of tea, where H represents hide and t represents time for of all that's been tossed. Um, and we need to figure out First off, How high is it? At T equals one, in other words, after one second. So we want to find a job one. So that means we'll substitute one in for teas of 16 times one squared plus 32 times one, and that equals negative 16 and positive 32. So that's 16 feet. That's age of one. We also need to determine what's H of three halfs. So after 1.5 seconds, how highs it's that will do the same process only this time will substitute in three halves less 32 times three halves that equal. Let's see. Three hubs, 9/4 negative for so this is negative. 36 32 divided by 2 16 plus three is 48 48 months. 36 is 12. So this is 12 feet. So part A. These are our two answers that one second at 16 feet and at 1.5 seconds it's at 12 feet. All right, so then next up we need to determine what is the maximum kite and this is a quadratic. Let's take a look right here. We know that it's con cave down. So if we were to graph it, we'd see a problem like this and the maximum high. It's going to be the Wyke ordinate of the Vertex. So we want for text out Why so vortex dot acts. The X coordinate is equal to negative. Be over to a where B is the coefficients of the T term and a physical efficient of the T square term. So this is gonna equal negative. B is positive. 32 two times a is two times negative. 16. So that's negative. 32 over negative 32 equals one. So h of one is the maximum height. So the time is one second and her text Why is H one which we already figured out a 16 feet? So that's our part. B answer is the maximum height. It's 16 feet and it happens at one second. T equals one second. And then the third thing we wanted to do is, um what is the time when the height is equal to seven feet so what we're going to do there. And that's part C. So we're going to do to get that is set h of t equal to seven and solve it. So we're gonna have seven equals. Negative 16 t squared less 32 t. So why don't we solve it? But the everything on one side of the equal sign and zero on the other and that we have a couple options. So let's add 16 T square to both sides. Subtract 32 t add seven and set it equal to zero. Now we could either use the quadratic formula or we could try to factor it. Um, six of 1/2 dozen of the other. If you're really good at factoring, factoring, it does factor my dragic formula, though no guessing Jack involved. So let's go that route So quadratic formula this negative be So let's just remind ourselves this right here is be and this is a and this is C so quadratic formula is negative. Be so negative. Negative. 32 plus or minus the square root of B squared. So negative 32 squared minus four times a times. See over two times a And although that looks intimidating. It's actually gonna work out just fine because 32 squared minus four times 16 times seven and take the square root of that is a nice entered your number, so we're gonna get 32 plus or minus 24 over 32 so that's pretty easy. 32 plus 24 is 56 and 56. Divided by 32 is one and 3/4 so it's gonna be 1.75 Or, by the way, that can also be written 7/4 right weaken right 7/4 and the other one is 30 to minus 24 which is eight. Divided by 32 is a quarter's 0.25 or you could have written it one four. So that's the two times at which the height equals seven feet. And by the way, we need to put our units in there. It's in seconds because units matter

The ball is thrown in the air. And if the path of the ball follows the given function, so we need to find the initial height, the max high. And when does it hit the ground? So the initial high is when T. Is equal to zero. So we substitute zero and 40. We get negative four times they had a 4.90 square plus 24 times zero plus eight. So our initial height is At 08, so that corresponds to eight meters. Yeah. Is our initial height are maximum height occurs at our vertex. We need to find our access to symmetry, so X. X is equal to negative B over two. A. Actually it should be T. So we have T. Is equal to negative B over two. A. So is the number in front of X squared B. Is the number in front of X of T. So we have negative 24 divided by two times negative. 4.9, -24, divided by two times negative 4.9 Gives us that the time it takes is approximately equal to 2.45 seconds. So now we're going to replace the 2.45 into the age of T function. So we have the age function evaluated at 2.45 seconds. So we have negative 4.9 times 2.45 squared plus 24 times 2.45 plus eight. So negative 4.9 times 2.45 squared plus 24 times 2.45 plus eight. We get that the maximum height is approximately equal to 37 point four meters. So now we're going to find when it hits the ground. So that means we need to find our X intercept. So we're going to use the quadratic formula. So X. Is equal to actually T T. Is equal to negative B plus or minus the square root, B squared minus four times a times C over two times A. So we get negative 24 plus minus square root, 24 squared -4 times negative 4.9 Times eight. Cassie is eight over two times -4.9. So the first answer we get by doing subtracting some negative 24 negative 24. We're gonna add the square root of 24 squared -4 times negative 4.9 times eight, Divided by two times negative 4.9. So when you add you get AT is equal to negative 3.313, which is not a valid answer. So now we're going to subtract so negative 20 for minus the square root of 24 squared minus four times negative 4.9 times eight, divided by two times negative 4.9 we get T is approximately equal to 5.2, So that is the valid answers so it hits the ground in 5.2 seconds.

We're given the height function for a baseball that starts out five foot above the ground when it's hit and were asked to figure how long it's in the air. Well, we know that when the height gets back down to zero, it's on the ground again. We weren't told a specific method to solve this one with. So I am going to use the quadratic formula we're trying to figure when the highest zero we have our equation. We know the quadratic formula in this case, T equals negative B plus or minus the square root of B squared, minus four a c all over twice. A. So if we fill those in, we'll have t equals negative. 76 plus or minus the square root of 76 squared minus four times negative 16 times five all over negative 32. We have negative 76 plus or minus. Inside the square root 76 squared is 5776 four times negative 16 times five would be negative, 320. But since we're subtracting that, we will put minus negative 3 20 or plus positive. So now we have t equals negative 76 plus or minus the square root of 6096 over negative 32. The square of seven or 36,096 is approximately 78 0.7 So I'm gonna fill that in now. We're gonna find two solutions here, but one of the answers is a negative number. It can't be a negative time, so we only want to look at the positive time. The positive time is going to come when we subtract the 78 0.7 So I have negative 76 spine A 78.7. That's negative. 1 54 0.7 Divided by 32. Show me divided by a negative 32 which, if I fill that in 1 54.7 negative divided by 32 negative. The time I get is four point 815 seconds

Okay, so first question asks what the initial height of the ball is. So initially means Time he called zero. So I'll have to do is substitute in T equals 09 of getting H equals four and that's going to be in feet. So that's the initial height of the ball just before being released. Okay, Now the second question Asks, when is the ball 18 ft above the ground? So h equals 18. I have negative 16 t square plus 30 t minus 14. He called zero and I can rewrite this A 16 t square minus 30 minus 30 T Plus 14 equals zero. And so if I sold for this I can say T equals minus B 30 plus or minus square root of B square 30 square minus four. Mm. See uh huh. To A if I solve this, I end up getting 0.875 seconds and one second. So when the ball's thrown it will follow a path. Something like this. Right? And so if on the vertical axis I have height On the horizontal axis, I have time. If this marks 18 ft, there's going to be two points when it's going to be at 18 ft. One is when it's traveling upwards and when one when it's traveling downwards, so when it's traveling upwards It reaches 18 ft at time 0.875 seconds. And when traveling downwards it reaches Uh it comes back to the same height of 18 ft at 1/2. And so finally, the last question asks, how long does it take for the ball to hit the ground? So see I just substitute each equal zero or rather? So yeah, the player throws the ball like this travels like this and then falls here. This is the ground, This is where the player launches it at. So that we calculated in the first part, that's four ft And we want to know how long it takes to reach this point over here. So that is zero ft. So all I have to do is substitute at H equals zero, so H equals zero. What I'm going to have is 16 T square minus 30 T. Yeah minus four. He called zero. And then I just saw for this. So this can resolve by saying T equals 30 plus or minus 30 square minus four times 16 times minus four. Yeah, Over two times 16. And this gives me T equals -0.1, seconds and equals two seconds. I can eliminate this negative because negative time does not make sense. So the answer is two seconds.


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