the-height-of-a-ball-that-is-thrown-directly-upward-from-a-point-200-feet-above-the-ground-with-an-i

Name: the height of a ball that is thrown directly upward from a point 200 feet above the ground with an i
Uploaded: 2020-05-09
Duration: 4 min 38 s
Channel: Numerade
Description: The height of a ball that is thrown directly upward from a point 200 feet above the ground with an initial velocity of 40 feet per second is given by $h(t)=-16 t^{2}+40 t+200,$ where $t$ is the amount of time elapsed since the ball was thrown; $t$ is in seconds and $h(t)$ is in feet. For what values of $t$ will the height of the ball be below 100 feet?

Question

The height of a ball that is thrown directly upward from a point 200 feet above the ground with an initial velocity of 40 feet per second is given by $h(t)=-16 t^{2}+40 t+200,$ where $t$ is the amount of time elapsed since the ball was thrown; $t$ is in seconds and $h(t)$ is in feet. For what values of $t$ will the height of the ball be below 100 feet?

Aamir Mithaiwala · Accepted Answer

all upward and we want to know at what time is a below a certain certain. So we want to know where this ball is below 100 feet if we threw it at 200 feet. And so now we are given a a height function. But we want to know where that height is less than 100. And so if we have the quadratic or we have the function of the height, we want to know where it&#x27;s less than 100. We can figure out where where we can. We can manipulate that and say OK, so we have we have negative 16 t squared negative 16. You have 16 t squared plus 40 t and then since we subtracted 100 to get it to make make the height lessons here is that we have plus ah, 100 is less than zero feet. And so now we can solve using the quadratic formula. So we have negative 40 plus or minus plus or minus the square root of B squared, which is 16. It ISS 16 100 minus for times negative. 16 4 times negative 16 times 100. So that&#x27;s plus 6400 plus 60 or 100 Ah, 6400 over two times Negative 16. Negative, 30 to. And so if we were to solve for this, we would get we would get the root of the square root of 1600 plus 6400 is the same thing as saying is the same thing of single the square of 8000. So the square root of 8000 is 89.442 But if we say negative for you, plus this value. So if we were to softens, we can we can figure out the values of T. It&#x27;s a let&#x27;s a T is equal to well, we have negative 40 plus 89.442 which is a square of 8000 that and then provided by negative 32. That leaves us with negative 1.5 negative 1.5 and the other instance in which case we subtract negative 40 minus that value negative. 40 minus the square root of 8000 we&#x27;re left with. We&#x27;re left with 4.4 so four point zero for Or let&#x27;s keep it at four points here or four seconds, so it&#x27;s round to the nearest whole number or point. But let&#x27;s round two, the newer stencil. So we have We have two instances. We have a negative time and we have a positive time. So we already know that in real world applications, negative time doesn&#x27;t really we don&#x27;t worry about that. So, missus is we have t is equal to four. And so if we were to draw a graph of this, we have at at time T equals zero Time T equals zero were at some value 200 and we go down. And so we want to know we want to know Where is this? Where is this? Less than some value. And so if that occurs at four seconds, T equals four seconds. We know that when t that when t is greater than for that is when our height will be less than 100 feet

Daniel Nolan · Answer

So in this question, a ball is being Donald puts on its height in feet after. See seconds is given by this equation. Why? Because 40 t minus 16 t squared, and it&#x27;s asking us to find the maximum height attained by the ball. So why is the max is the height of the ball after tea Seconds to find the maximum height, we just need to find the maximum value of this equation, and then they&#x27;ll give us the answer. So to find the maximum value, I&#x27;m just going to use on a equation for the I&#x27;m to find the that the tea co ordinate for the Vertex on. Then I&#x27;m gonna send that back into the equation. Allow give, Millie. Now, give me my maximum value in feet. Okay, So hit. I&#x27;m gonna use this equation. It&#x27;s gonna be minus b over to a and I&#x27;ll give up me on the team. Coordinate off the vertex. So minus B here is minus 40 on dhe to a is minus 32. Okay, just gonna simplify that with my calculator. And that gives five over four. So if I sold five over four back into the equation now, I&#x27;ll get the maximum height attained by the balls of your feet. That&#x27;s gonna be 40. Type by five over four, minus 16. Touched my five over for all squared. Looking right into my calculator. I get it 25 that&#x27;s in feet. And that is the maximum height attained by the bar. Okay, thank you very much for listening. If you have any questions about that, you can email me at Daniel D. J. P. H. Not known, uh, at gmail dot com. There we go. Thank you very much. Listening and wish you the best of the best for your studies.

Amrita Bhasin · Answer

this question instructs us to find the maximum height obtained by the ball. What we know is that we have the function after of tea is 40 T minus 16 T square. This function has a maximum value because R is negative. Therefore, we can use the formula negative be over to eight solved. We know our B is 40 or a is negative. 16. This gives r T as five divided by four plug after five divided by four n Again, This is into the original function we&#x27;re solving for our second co ordinate tweet. We end up with 25 because this is a word problem. Don&#x27;t forget your unit. 25 feet is the

Bobby Barnes · Answer

we&#x27;re told a ball is thrown in the air with velocity afford a few per second, and its height can be given by the equation. Why is even 40 team minus 16 t squared? You want to find a velocity when time is acquitted, too. Now the thing will need to recall from physics is that velocity is equal to the derivative or the rate of change of our displacement. So we normally right, this is like Delta lie or don&#x27;t t. But in this case, what we&#x27;re really finding is the derivative of this function active. So if we were to call this function, f a t were wanting to find what is f prime of two. So what we would do for that is the limit as tr purchased two o f t minus Uh uh too All over. T minus two. Now we need to figure out wood after two is so let&#x27;s see. Effort to is going to be so it B a D minus 16 times two squared, which is or which is going to be 64 and then subtracting those should give us 16. So we&#x27;re gonna plug 16 in perfect too. So we&#x27;re gonna have the limit as t approaches to of so effort ease the puncture were given, and I&#x27;m just going to rewrite it like this. Where e. T squared is first. That may have minus 16 all over tea and minus two. Now, if we were to just plug this in directly, we would get 0/0, and I don&#x27;t know what that means. So what we can try to do is defied these two functions to see if they cancel. And I&#x27;m going to do this with synthetic division off on the side here. So it&#x27;d be too on the outside because that&#x27;s the root of T minus two. Read a negative 16 40 negative 16. So dropping the 16 down to that give us negative 32 which would give eight and then two times eight is 16 0 So that means this would be negative 16 x plus, not expert t t plus eight. Which means we can rewrite this us the limit as t approaches to, uh, negative 16 t plus eight. And now we can go ahead and plug to win directly into this, which would give us so negative 16 times two special. Let me just go ahead and write it out first. That&#x27;s gonna be negative. 32 plus eight or negative. 24. And the units with this would be feet per second. So this would be our velocity after two seconds.

Andrea Lee · Answer

the problem. You see Equation 40 team in the 16 T Square as well as the initial velocity of 40 feet per second. The problem asked to find the velocity AT T was too. Which basically means finding the isn&#x27;t thing is really changed. So we plugged in the equation. Uh, institutions may change for one TZ with two we&#x27;ll get be of two is equal to the limit this t goes to of effort T minus effort to over T minus two. Now you have the final one of two is equal to. So if you&#x27;re putting two into the equation, OK, 40 terms to 16 this equal to a T minus 64 which gives you 16. You plug it back into the equation limit t to you. Plug in the equation. 40 T minus 16 t squared, minus 16. No routine rightist, too. And now you factor out the negative eight to help simplify when there is a TV host to negative p times to t square five t close to oh, teen. Now you factor out to t square minus five t plus two a woman t ghost to he to t minus one SWAT team minus two. All their teams to and I can cancel up t minus two. This gets you a little. It has t goes with to open native eight to t minus once. And now, since there&#x27;s nothing else, you&#x27;ve been simplified. What? You to just plug in the to the equation. So too minus one. Then this gets you negative. Eight times. Four minutes? Well, just three. Get you native. 24. He cursed sick.

Nick Johnson · Answer

Okay, so we have the, um the bowl is, um, about the ground is giving my dysfunction. Here s a T is equal to native 16 t squared, plus 160 t. So when the bottle is 400 feet while from the ground that our function as of T is equal to 400 we just set SRT here equal to 400. Okay. And then we can go ahead, and we can, um well, just set this equals zero. We have negative 16 t squared, plus 160 teeth and then minus 400 is equal to zero. Okay, so, um, let&#x27;s go ahead. And this Make the coefficient on rt square positive by basically this multiplying everything by native one. So we have positive. 16 t squared, minus 160 t plus 400 is well, then still equal to zero. Okay, so now we have a quadratic equation here in standard form, which we can go ahead, um and well, let&#x27;s have a different way to solve it. Um, well, first of all, that&#x27;s actually condemn Vied. Everything here looks like by 16 to make the numbers a little bit smaller. If we do that, we end up with just t square minus 10 t. Um, and then while plus 25 is equal to zero. Okay, So? Well, if we can go ahead. You know, in factor is probably the first thing we&#x27;re gonna try and do, and it looks like it&#x27;s been a factor. Probably pretty nice. So T squared, minus 10 T plus 25. This is actually, um t minus five times T hopes times T minus five. Right. I always go ahead and check this by distributing, and we end up with well, t squared minus five T minus five teas and minus 10 t and then minus five to minus. Fine is a plus 25. So T minus five times two minus five. Well, that&#x27;s the same thing as t minus five. Whole quantity square is equal to zero. Um, so we see here. Well, the only does it makes this equal to zero. Would be would be five. Right? So So t is equal to five. So, um, the ball will reach 400 feet. At T is equal to buy seconds. Okay. And then we&#x27;ll describing this. And where&#x27;s they say um, we can say that, Um well, the ball, the ball, um, reaches, um, its maximum height. So it&#x27;s not so. Which is It&#x27;s max height. Um, aunt five seconds. Um, because, well, this is the only time that it reaches 400 feet. Right? The Onley value of tea that makes his truth is take was five. So it reaches its max height at five seconds because, um, this is the long time. So this is Theo. The only time, um, that it reaches 40 feet. We use 400 there.

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Problem 42 Hard Difficulty

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\(t\) \(\geq\) \(4\) seconds

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