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A balloon upward velocity is [Ot-t' fcet per minute minutcs after It IS rcleased from fcct above the ground:. A) Find the position function of thc balloon_ 18W...

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A balloon upward velocity is [Ot-t' fcet per minute minutcs after It IS rcleased from fcct above the ground:. A) Find the position function of thc balloon_ 18What is its acceleration afler 5 minutes?The slope of a curve at any point is the equation of the curve:=Vrtxtl and the curve has the point (9, 2). Find

A balloon upward velocity is [Ot-t' fcet per minute minutcs after It IS rcleased from fcct above the ground:. A) Find the position function of thc balloon_ 18 What is its acceleration afler 5 minutes? The slope of a curve at any point is the equation of the curve: =Vrtxtl and the curve has the point (9, 2). Find



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Vertical Motion A water balloon is thrown upward from the top of an 80 -foot building with a velocity of 64 feet per second. The height or displacement $s$ (in feet) of the balloon can be modeled by the position function $s(t)=-16 t^{2}+64 t+80,$ where $t$ is the time in seconds from when it was thrown.
(a) Find a formula for the instantaneous rate of change of the balloon.
(b) Find the average rate of change of the balloon after the first three seconds of flight. Explain your results.
(c) Find the time at which the balloon reaches its maximum height. Explain your method.
(d) Velocity is given by the derivative of the position function. Find the velocity of the balloon as it impacts the ground.
(e) Use a graphing utility to graph the model and verify your results for parts (a)-

Hello. We have been waiting 30 number question in which hotter Bolognese rising vertically from the ground and high days given by the model half A T squared plus half T feet. What? He is between 0 to 60 seconds. So problem number A party body, the height of the balloon at the end of 40 seconds. So we need to calculate at 40. So let us plug in the value off T as 40 year half 40 Squire plus one by two into 40. 1600 by 2. 800 plus 20. That is 820 ft. This is the height at the end off 40 seconds. Now, Part B, what is the average velocity off the balloon between T called zero to take hold 40. So every velocity will be equal. Tour at 40 minus at zero by 40 minus zero as 4820 months. Zero will be zero by 48 20 by 40. 20.5. Okay, so every velocity is 20.5 feet a second. Okay. Part C says Bart ccs, what is the velocity of the balloon out in the 42nd? So we need to get calculate the velocity. So let us differentiate at with respect to t which will be equal toe two into one by two. T plus one by two That is deep lis one by two. So we have to find it is the last year at the end off 40 seconds. So V 40 will be equal toe 40 plus one by two That is 40.5 feet per second. So these are dancers. Thank you so much.

So we're given this problem in it states that a water balloon a drop from a window off all a distance of s equal 16 t squared feed during the 1st 2 seconds. Find about the loons. A average velocity in the 1st 3 seconds of this. Installing an instantaneous velocity. Remember, Average velocity one. Uh, so let's do a We have to find the balloons. Average velocity during the 1st 2 seconds. Average velocity, average velocity. He's gonna be equal to distance traveled in this case or distance, fallen over time taken. And so in this case, are distant. Fallen during the 1st 3 seconds will be given by us off three. So as off three divided by the time and Tokyo, we want to find the average warned during the 1st 3 seconds. So the time it took would be three seconds. So it's gonna be ass off three over three. Wait. So as off three over three is gonna be equal to Well, let's do I's gonna be ableto 1 44 over three. So 1 44 over three, and that is gonna give us 48. And it's really important to put units. So we are going, our units are gonna be feet per second. Uh, and though that's average velocity and let's do the next part, I'll just put another image of the thing. Problem. Let's do number. Be want to find the instantaneous velocity at Teeples. So right now we know that as softy over here gives us the distance forward, right? If we want to know the instantaneous velocity A T equals three So instantaneous Velocity Atika ls three would be equal to the derivative. Off s the function. As you call it s safety At T equals three. So it would be s, uh, as prime off three. So that's what we wanted. The instantaneous value AT T cool story, um, is gonna be equal to is gonna be that since you're in a row so we don't tree cool signs is gonna be as prime of three. That's gonna be our instantaneous velocity, because as prime as prime of tea will give us, it will be the derivative off herself, t, and that's gonna tell us the instantaneous rate of S f t. At any given time, any after, um, at any given time, any given t value and so we want to find the instantaneous velocity off. Um, the ball, the the ball, the balloons, instantaneous velocity and tickles Do therefore will take the derivative off the I guess you could call this the position from which gives us the total distance phone after tea seconds. And so that is how we're gonna find her average velocity by taking the derivative of S f T. Which will tell us which tells us the distance Volin after three seconds. And so, if, as is equal to 16 t squared as prime off T is gonna be equal to 32 tea and all I did year was used the power. Um uh, evaluate the derivative S O s prime of thes 32 teeth. And now remember, I said, are instantaneous velocity will get that by doing as prime of three. And as prime of three is gonna be full of 32 times three is inimical and 96 feet per second. Remember, always put units 96 feet per second and so that right over there is our instant balloons. Instantaneous velocity. Again. What we did is we took this function over here. This function does is the total distance phone after the distance phone after three seconds and we took the derivative of that and that if we take the derivative of that, we get as prime of three in as prime. If you will tell us the rate of change of S F T at any given t value and that's gonna tell us or instantaneous velocity at teak was doing that function is the key to finding the instantaneous velocity a tickle street. So we find the function, and then we find the instantaneous rate of change off a softie at Teke Oh, story. And that's gonna give us the instantaneous velocity off the balloon. A T equals three and kind of sounds confusing, but on if you need to re watch this video a couple of times, it would be recommended. But, um, once you get the hang of these type of problems, it's really straightforward as to what you wanted. Um and so, yeah, it's 96 feet per second and I hold that's helpful

Discussion. The balloon has an exploration A. B. And the speed at which the ball is strong is B. Yeah. So when the blue ball is thrown outside the balloon at that instant, the ball has the initial spirit of the balloon that is baby. But the acceleration of the board is along the normal direction and that is equal to the gravity. Still exploration of uh due to the earth. So for part A, the acceleration of the ball is in question minus G. The acceleration is -9.8. We dispose 2nd square. This is a long by direction. Now, let's say initially when the ball was released, it has a height. Why not? So for part B. From the equation, the final velocity squared equals to Misha village city square. Yes, to A into why for the balloon balloon, the Why not is equal soon since the initial Swedish zero, you can neglect this one, the final speed of the balloon square upon to into the acceleration of the balloon. And from this you find the position of the balloon when the ball was thrown. As he questioned The final spirits, 15 m for a second Square upon to into the exploration of the world is three m/s squared. The value of the why not is 37 0.5 m. Well, in this situation, the ball will follow a cinematic path So we can write the position way of the ball two times the initial question. Why now? Plus the initial velocity of the ball. That is not T plus half of the exploration of the ball. Airway into T square. Okay. And final position when the ball hits the ground by will become zero, so we can neglect this. So zero m is equals to 37.5 m. Let's 15 m. post again into T. First Half of -9.80. We just for a second square into the east. And from this, we get a quality question that is 4.9 D square minus 15 T Uh -37.5. That is equal to see if he saw this. These equations, We find the positive solution for the situation is T equals to 4.7 seconds. So after t close to 4.7 seconds, the ball with will hit the ground.

A ball thrown vertically in the air from ground level has height 18 t minus 9.8 E. Squared. And that's in meters and T. Is in seconds. So, they want us to find the average height and the average speed. So that's two things of the ball over the time interval from when it is released to when it hits the ground. All right. So first we have to find out what the time interval is. So when it's released it's at level zero and when it hits the ground its height zero. So all we have to do is solve this zero equals 18 t -9.8 He squared background of T. So T is zero and T is 18 Over 9.8. Okay. T0 that's when it's released 18. Dubai by 9.8 is when it hits the ground, 18 divided by what's 18 divided by 9.8. Okay, that's a messy thing. So, I'm just gonna leave it as 18 over 9.8. Okay. To find the average height. eight. What's H. Bar? It's one over the time interval Which is 18 over 9.8 0 integral 0 to 18 over 9.8 18 T -9.8 E. Squared detail. So think of this as the area divided by the length. And that will give us the average height. So this gives us 9.8 over 18 18 T squared over two -9.8 T cubed over three From 0 to 18 over 9.8 nine porn aid over 18 18/2 times 18 squared over 9.8 squared minus nine point 8/3 times 18 cubed Over 9.8 cubed. Okay so this isn't really helping me as it. I was hoping that. Oh yeah okay well I can do a little bit here. 9.8 over 18. I'm gonna factor in 18 squared over 9.8 squared out. So that leaves me 9 -9.8 for three times 18 over 9.8. Okay. Those cancel that cancels and leave me six. Okay this is okay. So now I got three in the in there and then one of these cancel one of these cancels. So 18 over 9.8 Times three. OK, that's not terrible. Okay. Or you could just do the whole thing with the decimal? Okay, 18 times three divided by 9.8. 5.5 I get for the average height 5.5. Does that make sense? 5.5 m. Um Yeah yeah that could make sense. Now they want the average speed so I'm gonna have to take the derivative to find the velocity. Okay, HRT remember was 18 T minus 9.8 t squared yes So the velocity will be 18 Martinez 19.60. Okay and remember that speed is just the absolute value of the velocity so oh that's a drink. Okay so uh speed absolute value of the velocity. Okay so that'll be 18 -9.6 T 19.6 T. When 18 -19.6 is positive. And that will be the opposite of that When that's negative. So this is from zero two T equals 18 over 19.6. And this will be after that 1918 19.6 Until it hits the ground at 18 9.8 because it's a problem it goes up for half the time and then then it goes back down for the other half of the time. Okay so to find the speed you have to integrate twice. Okay first on the way up and then on the way down. Okay. But they're really just the same so we'll just do twice the first integral. Okay, so the average speed will be won over the interval twice. The integral 0 18 to 19.6 18 -19.60 T. T. So then this flips around and then we have this to here. So we have 19.6 over 18 18 t minus 19.60 squared over two From 0 to 18 over 19.6 19 point 6/18. I'm gonna factor a T out. So I had leave me 18 -19.6 over to Times 18 Over 19.6. Okay, so those will cancel and then I'll have 18 -9 Which will give me nine. And those all canceled. So I get nine is the average speed nine m/s.


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