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4.A flaming arrow is shot into the air to mark the beginning of a circus act. The height, in metres, of the after seconds is represented byhut) 49" + 28,50+2De...

Question

4.A flaming arrow is shot into the air to mark the beginning of a circus act. The height, in metres, of the after seconds is represented byhut) 49" + 28,50+2Determine the velocity and acceleration of the arrow after 4 seconds the arrow speeding up Or slowing down? Use derivatives to determine when the arrow reaches its maximum height; What is the maximum helght? How long does it take for the arrow to hit the ground? At what velocity does it hit the ground? Sketch slt), v(t) and alt):

4.A flaming arrow is shot into the air to mark the beginning of a circus act. The height, in metres, of the after seconds is represented by hut) 49" + 28,50+2 Determine the velocity and acceleration of the arrow after 4 seconds the arrow speeding up Or slowing down? Use derivatives to determine when the arrow reaches its maximum height; What is the maximum helght? How long does it take for the arrow to hit the ground? At what velocity does it hit the ground? Sketch slt), v(t) and alt):



Answers

The height (in feet) of a projectile shot vertically upward from a point 6 ft above ground level is given by $s(t)=-16 t^{2}+48 t+6,0 \leq t \leq T,$ where $T$ is the time the projectile hits the ground. See FIGURE 4.1 .8 . (a) Determine the time interval for which $v>0$ and the time interval for which $v<0$. (b) Find the maximum height attained by the projectile.

This function S. Which gives us the displacement of this projectile. And so we want to find the velocity function V. So V. Is equal to the limit. His h. goes to zero of our function S. Of T. Plus H. So we plug in T. Plus H in for tea. So we have 44 times T plus H -4.9 times T. Plus H squared. And then we minus S. Of T. So ar minus ng 44 t -4.9 T squared. And then this is all divided by H. So what I'm gonna do now is distribute this 44 expand this factor here as well as distribute this negative sign. So this is going to be equal to the limit. It's a church goes to zero 44 T. Plus 44 h minus 49 times T squared plus two times H. Times T. Plus H. Squared. And then we have minus 40 40. And then we have to distribute this minus sign. So this becomes plus 4.9 t squared. And this is all divided by H. So now we can get rid of this positive 4040 with this negative 4040. And so this would be equal to the limit As each goes to zero of 44 h. And then I'm going to distribute this negative 4.9. So we have minus 4.9 times T squared And then -4.9 times two is equal to 9.8. So the 3 -9.8 Times H. Times T. And then we have -49 Times H. Squared. And then we have lastly this plus 4.9 times T squared and this is all divided by H. And so now we can actually get rid of this factor here 4.9 times T squared, The negative 4.9 times T squared with this positive 4.9 times T squared. And so we have this is equal to the limit As a church goes to zero of 44 h -9.8 H. Times T. And then -4.9 times h. Square. And this is all divided by H. So I can factor out an age now. So this is equal to the limit H goes to zero H. Times 44 -9.8 times T -49 times h. And this is all divided by H. And so now we can cancel out these two ages. And so we're left with the limit H. goes to zero 44 -980 -4.9 H. And we can let h equals zero. Now since we no longer have just h. And the denominator, so we won't be dividing by zero. And so this will be equal to 44 minus 9.8 T. And then 4.9 times H one H. Is equal to zero is just zero. So this is equal to our velocity function. And now we want to figure out at what time M. T. Is our velocity going to be equal to zero? So we're gonna set zero equal to this equation and solve for T. So we have zero is equal to 44 -9.8 times t. You can add the 9.8 T. Over. So we have 9.8 T. is equal to 44. So then T. is equal to 44 divided by 98. And so if we plug this into a calculator, 44 divided by 9.8 is equal to about four point About 4.5 seconds. So it takes about 4.5 seconds for this projectile to have a velocity equal to zero.

According to the question, we're given the equation for the height off the ball as Etch Off T is equal to re not. T minus 16 T Square, where a tuft is the height that the board reaches the knot is the initial velocity with which it is released, and T is the time in seconds were also given that the initial velocity with which the boys shot is 64 foot for a second, so we have to find the time in which it hits the ground. So when it hits the ground, hft will automatically become zero on me not is already given a 64 so this given equation will transform into let's number. This equation is one. So according to the question one will become zero is equal to 60 40 minus 16 T square, which implies that minus 16 t squared plus 60 40 is equal to zero, which implies that 16 T square, minus 60 40 will be equal to zero, which implies that if he takes extent common fact rising, so then we'll get T minus four is equal to zero. Now, from this step, what can we say? Which in place I'd owe 16 T is equal to zero or T minus four is equal to zero. So if 16 t is equal to zero than this implies that he's equal to zero. If t minus four is equal to zero than it implies, that is equal to four. Thought of zero and four we take. T is equal to four as the time. So what can we say? Therefore at a time, Deke will do four seconds. The bull hits the ground when it is released, with the initial velocity off 64 40%.

Alright. Looking at this problem here, we've got the height of something being launched in the air. And then we were also given Intergraph now part a tells us to use their graph to estimate how long this object is in the air. Well, we start off here at time T equals zero notice over here. That point right there which is a time T equals for its height, is also zero there. Because if you look at this the Y axis here, that represents your height and this was whenever hits the ground again and so we expect it to be in the air about four seconds, about four seconds now be tells us to do it. Algebraic lee. So it's going to hit the ground again when it's hide a zero. And so it actually solve this rhythm, putting in zero for age of tea and then solve this for T's. We have zero equals negative 16 t squared plus 60 40 and then we have a quadratic to solve. And so we're going to solve this by factoring to pull out a negative 16 t That leaves you with T minus four on the inside. And so using the zero product property that's tells us native 16 t zero or T minus 4 to 0 if name search team T A zero that tells us tea is zero. It just means it's on the ground whenever you launch it. All right, so that doesn't tell you how long it's actually in the air. This other part, though, does your tea is four foot 64 seconds before it hits the ground again. And so that's what we should have on beat. I don't see says to use their draft s maiden win reaches its maximum height, so I compared the graph. What time does it look like it is whenever it reaches this peak up here? Well, it's like it's at about two seconds. So we expected to reach its peak at about two seconds. Yes. We're going to say about two seconds now. When do we actually want to find an algebraic Lee? So we have a job t his name of 16 t squared loss of 64 teams. Now, if you think about this here, this is a problem, and so probably is gonna regions. But a just point at its vertex. And so here. We just need to find the X coordinate or, in this case, the T coordinate of the Vertex. So the T coordinate of the vortex performing was negative. Be over to a So that's negative. 64/2 times Negative. 16 a 64. Don't be negative. 16 during this is his nature of 64. Over. Negative 32. It would be positive too. So it'll take it two seconds before it reaches its maximum height.

The given question is accessing quotes to be no toe the advice because to be north Ruto B minus 16 T square we'll be notice given us 400 feet per second in the motion It is giving the president honest thrown head a attending for different bigly. So from the information we can conclude these are questions So many in a part We need to find the timing then near Esten valid here to step down. So if it hits the ground we know that this part must be zero This value must 30 because it will start at zero and with zero So we can say like this the graph local like this from here it will start and from here it will end. So if you see then by value here in zero. So if this is zero people get we know why two times t minus 16 80 square is equal to zero We not value is 400 Let's take one tea common that we will get for trade route Dubai to minus 16 The quotes 201 Cancer We will get Daisy Question zero debt indicates this position the other position We will get then sick. This this d there because And it becomes zero. So we will get 400 group toe by toe minus 16 d is a quote stoop zero that will give us b is equal stole toe red rule toe by 16 or tea is Cuba unwto approximately close to 17.7 2nd in second part. They are asking for the range this range. So go find that this strange. We can substitute this value off, be in this in question. Then we will get range is in questo X is equal stow. We know Roto two times t and that will be equals to be not the new 400. I don't know why do and duty at 17.7, that value comes out, do we? 5000 feet for the sea part. They are asking the value high. Yes, every do this will do to find this value. We know there at the highest. Only knew the time will be half so we can substitute their value in this equation half off key and get the answer. So let's do it here. So why is it wants to We know when you 400. So I go t is 17.7, developed by two minus 16 into 17.7. Divided by holy square. If you can clear, this will comes out to be 50 feet. So this is the solution.


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