height-of-a-ball-if-a-ball-is-thrown-directly-upward-with-a-velocity-of-40-mathrmft-mathrms-its-he-4

Name: height of a ball if a ball is thrown directly upward with a velocity of 40 mathrmft mathrms its he 4
Uploaded: 2020-06-03
Duration: 50 s
Channel: Numerade
Description: Height of a Ball If a ball is thrown directly upward with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^{2}$. What is the maximum height attained by the ball?

Question

Height of a Ball If a ball is thrown directly upward with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^{2}$. What is the maximum height attained by the ball?

Amrita Bhasin · Accepted 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

Aamir Mithaiwala · 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

John Irizar · Answer

for our position were given the function s equals 80 T minus 16 t squared. Now, to find the maximum height that the bone reaches, that would be where our velocity is. Zero. In other words, it&#x27;s stops going upwards and starts coming back down. Now our velocity is the derivative of the position which would give us a T minus 32 t setting this equal to zero. We get negative 32 t is equal to negative 80 and T is equal to 2.5. So that gives us the time when the ball reaches its maximum height, not the maximum height. For that, we will have to take our 2.5 and put it back into our position function. Eso the position at time to 0.5 would be 80 times a 2.5 minus 16 times 2.5 squared. And evaluating this we get 100 feet is the maximum height that the ball reaches. No, I&#x27;m going to draw very roughly here the shape of our position and we know want to know. We know that this here is 100 fee. We want to know the velocity of the ball on its way up when it passes 96 feet and also on its way down. So let&#x27;s find out what time it reaches 96 feet. So I will take our position function 80 T minus 16 p squared and said it equal to 96 to find the time that it reaches that point. So I&#x27;ll bring everything over to the right hand side. I&#x27;ll have 16 t squared, minus 80 t plus 96. Now everything here is divisible by 16. So let&#x27;s go ahead and get rid of that factor. Well, t squared minus five t plus six. And this factors into T minus two and T minus three. So the two points occur and two seconds which would be on its way up and then at three seconds on its way down. Now let&#x27;s find our velocity at those two points. So the velocity at T equals two. I would give us K G minus 32 times two, which will give us 16 feet her second. The velocity AT T equals three would give us a D minus 32 times three, and this gives us negative 16 feed her second the negative, indicating that it&#x27;s coming back down

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.

Christine Anacker · Answer

All right, So we&#x27;re dealing with another word problem here about a ball that&#x27;s being thrown straight upwards from the top of a building. They give us the initial velocity, but it doesn&#x27;t really matter in this case because they already give us the equation so we don&#x27;t have to make it ourselves. So the function they give us is negative 16 t minus plus 32 t plus 48 and they want us to find the maximum height. So all we&#x27;re gonna do is we&#x27;re gonna look for the Vertex here, So I&#x27;ve got a rough sketch of what the graph is to find the Vertex you&#x27;re gonna go into second trace. This is a maximum because we&#x27;re looking for a maximum height and then just make sure that you&#x27;re doing your left and your right of the value and then hitting enter. So if I pull the actual graph up, it looks something a little bit like that and I can see that my highest point. My vertex is at one and 64. So once you do all that work, you should get a point of one and 64. That means you&#x27;re Max height ISS 64 feet, and that occurs at one second

Problem

Path of a Ball A ball is thrown across a playing …

Problem 51 Medium Difficulty

Height of a Ball If a ball is thrown directly upward with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^{2}$. What is the maximum height attained by the ball?

Answer

$25 \mathrm{ft}$

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