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18. If Fx)arccotdt, find F' (x).19. A ball is thrown vertically upward from the height of 9 ft with an initial velocity of 36ft/s. Use a(t) ~32ft/s? to find ve...

Question

18. If Fx)arccotdt, find F' (x).19. A ball is thrown vertically upward from the height of 9 ft with an initial velocity of 36ft/s. Use a(t) ~32ft/s? to find velocity and height functions_ b) How long will it take the ball to rise to its maximum height? c) What is the maximum height?

18. If Fx) arccot dt, find F' (x). 19. A ball is thrown vertically upward from the height of 9 ft with an initial velocity of 36ft/s. Use a(t) ~32ft/s? to find velocity and height functions_ b) How long will it take the ball to rise to its maximum height? c) What is the maximum height?



Answers

Height of a Ball The path of a punted football is
given by the function
$$f(x)=-\frac{16}{2025} x^{2}+\frac{9}{5} x+1.5$$
where $f(x)$ is the height (in feet) and $x$ is the horizontalal
distance (in feet) from the point at which the ball is
punted.
(a) How high is the ball when it is punted?
(b) What is the maximum height of the punt?
(c) How long is the punt?

You know this problem. I wanted to use the draft that we have been given to answer a few questions now. The graph represents the height of the law thrown. So the Y. Value is the height and then the X. Value or the T. Value there on that T. Access is the time that it has taken. Yeah here on it we want to know what is the height of the ball at two seconds. And so it just means find the value of why whenever you're X. Value is to find the value. Why when your X value is too. So that looks like it is at about 240 ft. Okay. Mhm. On be it says we want to know when will the height B 192 ft. And so find them why value of 192. And then find where on the graph find what T. Values give you a Y. Value of 1 92. There are two of them. And that would be at T. equals one. Okay. And the tear doubles five at both of those values. It's and see if we want to know during what time interval is the ball going up. So just look at this graph here. When is it going up? What goes up between two and three? That's just whatever that ball is going up And then it goes down after that. So from three. Mhm. The seven that's when it's going down. Do you want to know how high does the ball go? And when does it reach its maximum height? So you look for your peak there and the peak is at 256 ft. And then you also want your Y. Value that. I'm sorry, you also want your T. Value there and that would be at time T equals three seconds. That is a t. value at that height. Yeah. And then lastly, on e after how many seconds the ball hit the ground. And so you want to know when is it back down to the very base? When does it hit the T. Axis? Former graft, I wouldn't say it has the T. Axis won T. The second.

Yeah, we are even already function. Actually, Google minus you couldn't leave. Where? The best guarantee. That's six. It is expected in the problem. You find the maximum of one. The function and the blue party or region here excluded. Prevent over the wall in feet. How we knew loaded an acceptable was The function occurred at the Vertex after off the function at home. Okay, find the edge used. Blew A No. Maybe you have a new off A and B and a my 16 lead directly. I see it. Six. I would leave that to the formula he had And a quick look manage. Currently managed only. Yeah. You don't want to fight here? Uh, me. Do you get a 65 and you could find up that left a So what? Well, they will play in a given function as the little 0.65 I mean, the 65 that apparently the six, I think no, it simply fight you here. All right, Blue fight with the Knicks. Yes, The fight. Oh, my fight it. Did you look for them, huh? Really? And it is a little bit and I now the whole greeted maximum off blue okay and Exhale Night and the first time a the point 65 after it had ruled that the final lengthen it is.

Your problem. 11. So in this case, were given the height of the baseball which is given by the equation. The first party. We have to find the height when years one. So let's put years one over years we get minus 16. Prestigious do, which is nothing but 16 Onda. Next case we have a report a crime mystery over to so the height will be minus 16 times tree over two holes were plus 32 times three or two. So from this we get the value of high dressed well and Barbie There Acindar was the maximum height and that time vender reaches the maximum height is the quadratic equation. So for quadratic equation, the maximum point occurs when it is the A minus bi over the way. This means that minus off this will be 32 Will there is a value will be over two times minus 16. So that just gives the value one and br badgered the one we already form, which is 16. So the Heitor, despite its 16 this is the maximum and the last part asks a stark words that time What other times that one night a seven so This means that minus 16 p square prostitute t we have required to seven the skin for the video last 16. These where I'm Mr To Duty plus seven is equal to zero. We took everything or to the left, to the right. And then I think this can in fact rise. This converted tonight 16. He's going minus 2080 minus for 40 plus seven is equal to zero. That's take 40 old from here, so we get 40. This is 40 minus seven on that stick minus one. From here, we again get 40 minus seven is equal to zero. That's still 40 minus seven out again. So we get 40 minus seven times for B minus one. So from here, bacon into values off the one a seven or four and one is Baltimore for seven or four Meted will be for Transformers 4751% repair. And this is 0.2 faith. So these are the two values off

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


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