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Question

Because the displacement $s,$ velocity $v,$ and time $t$ of a moving object are related by $s=\int v d t$, it is possible to represent the change in displacement as an area. A rocket is launched such that its vertical velocity $v$ (in $\mathrm{km} / \mathrm{s}$ ) as a function of time $t$ (in s) is $v=1-0.01 \sqrt{2 t+1} .$ Find the change in vertical displacement from $t=10 \mathrm{s}$ to $t=100 \mathrm{s}$

Zulfiqar Ali · Accepted Answer

on the origin For X X relation along hex, we can get to any creature for X so X is equal to one divided by 48 times. R t to the r four. Well, inserting disagree gin in certain this into equation into equipment for why we get why is equal to Route 100 120 times 10 to the power three. A motive. My by one day body by for it most of my by t floor on debt views Ex elation along. Why is equal to 100 meter for a second square? Well, or Alexa relations then, uh, overall x relaciones then we have a is equal to root a X squared. Uh, let&#x27;s see a by square a musical to 25 square less Ah, 100 swim. So I thought Alexis relation is equal to 100 pre form 01 meter or second square

Surjit Tewari · Answer

question number 79. The acceleration function is equal. Okay. Minus B he to the power. How the ex elation has units or meter per second squad. So this constant a should also have, you know, adult meters per second squared. And since in the second be peeking through the powerhouse should also have you Nittel meters per second squared. But see, how has unit off second good A powerhouse. So the unit obi should be made that what? A second to the power. Bye bye. No, The power off this second and second, huh will make that total power off to and as the unit off B t to the power have will become meter per second misfire for the B. The velocity function is equal to integration Off the ex elation This is a call to it. Be less be d to the power three by two, reworded by three White too Last see if they put equals zero, then see will be equal to zero. So the value of the constant C is equal to zero So we can write be Tzeitel is equitable. It be thus it VT 00 to 80 0 Let&#x27;s be the Tzeitel to the power t by two upon three way too. Similarly, the position function is equal to the integration off the villa Skip function. This is a little it be is squared by two minus being he to the power bye bye to upon five by two in the three. By So plus, see, we can see that if eggs eagle, then the right hand side will become zero plus c. So here we can also remove the constant the rockets position as a function of Time T zero is get given. Us X t zero is equal to 80 0 It&#x27;s square by minus B C zero to the power if I buy two for in tow for by 15.

Hariprasad Annamalai · Answer

everyone. So what we have is the acceleration of a rocket that is governed by this equation shown here six plus 0.2 times D position and the units running meters per second squared over asked to solve for is the time at which the position is 100 m. Given that the initial velocity and position at T equals zero or zero, what we can do is used the relation that eight times DS is equal to P D. V. What we can do now is just plug in our acceleration that&#x27;s given to us for a and develop our limits as well. So the integral from zero to s of six plus 0.2 s ds is equal to the integral from zero to be Vdv carrying this integrate on the left and right sides gives us that six s plus 0.1 s squared is equal to one half V squared. We can see that when we saw for velocity in terms of physician that this is equal the square root of 12 s plus 0.2 s square. Now we consider DS is equal to v DT because velocity is equal to the derivative position with respect to time. So we just re arrange some terms. And now what? We can do this if integral from 0 to 100 which is the point in the position that we&#x27;re interested in In a grow over of DS over the square root of the block with square root, which is 12 ass plus 0.2 square is able to the integral of zero to t of DT. Carrying this integral out gives us that one over the square root of 0.2 times Ln of square root of 12 s. Plus, is there a 0.2 squared plus s time to square root of 0.2 plus 12 all over to Times Square of 0.2 Evaluated from 0 to 100 is equal to the time it takes for that rocket to reach an altitude 100 m. Now, plugging these limits and 100 and zero, we see that the time it takes is equal to 5.62 seconds. That&#x27;s a final answer

Anna Waldram · Answer

we&#x27;re given the equation for the position as safety of a rocket at Time T given in seconds when T is on the interval from 0 to 10 now, part a asks for the rate of change in position, and we were called. That rate of change is described by Slope, so we want thes slope of the tangent lines and this is given by the derivative. So for part A, we just want to take the derivative of us of tea. And so first will take the derivative of first term negative five t squared, and that gives us negative. 10 t Add the derivative of 40 T, which is 40 and derivative of 100 is zero. So that gives us part. A. Now, before we move on to the other parts, I think that doing part D next will actually be the most useful to us, so that will help us visualize the next parts were asked to graph the position from when tea is from 0 to 10 as well as the instantaneous velocity graph, which is given with this equation that we just found in part so we can do that using a graphing calculator And when we graph, we get these this one being the position graph and this next one would be the graph of the velocity when we put those into a graphing calculator. Now, this is going to help us with the next two parts for part B, we are asked, When is the instantaneous velocity zero? So what we&#x27;re asking for is we want to see this point where we cross the T access at zero velocity. This corresponds to this point the very talk on the graph where the rocket goes up and then it stops its velocity and starts to go downward. So in order to find this, we really just want to find where we&#x27;re crossing the T access in our velocity graph. And so all we have to do is set the velocity equation negative 10 t plus 40 equals zero. And so for teeth. And when we do that, we get t equals four seconds. So that is the time at which the instantaneous velocity is zero. Lastly, for part C, we want to know When does the instantaneous velocity have the greatest magnitude? So again, we&#x27;re looking at the graph of the velocity and when we use the word magnitude, we want the biggest number, whether it be positive or negative. And since that&#x27;s a linear graph, we can see that it&#x27;s having it has the greatest values. That&#x27;s the top here and bottom here. So I just want to check the endpoints and see what those are. We know that the top is CI equals zero. Bottom is t equals 10. So if we plug both of those into our velocity equation at zero, we get the velocity to be 30 and A T equals 10. We get it to be negative 60 and though it is negative, it has the larger magnitude. It&#x27;s the larger number, so that is going to be the time with the greatest magnitude of velocity. And so our answer is t equals 10 seconds.

Darshan Maheshwari · Answer

question on the 68. So we&#x27;re given a requisition off the position off a rocket when it is launched vertically. Airports. It is represented by S and S is equal to minus five p store rose 40 p. Messenger here we have to find the parties is that we have to find the rate of change of off this position. But retrospective pain just DS for duty, which is nothing but the velocity as well. So that will be delivered to off s with respect opinion so minus fifties were lost, 40 less hungry. So there&#x27;s a video of this will be administrated, a constant on off square with cruelty off 40 before piece of constant and they were to appease 100 is a monstrous was you? This will be minus 10 deepness for P. This is the velocity instantaneous velocity start over part B part being sees the venison velocity seed over being zero and then so velocity president and miss the time is the range of times between zero. So the zero means minus 20 plus 40 0 Let&#x27;s, uh, subtract for people of sites we have minus 10. India&#x27;s wings for people. Let&#x27;s divide minus 10 boards states. So we have DS minus 40 minus turn. This is nothing but for so I equal to force against the velocity zero, this is part B. Starting on the part sees parts, he say&#x27;s that Venice. It would also mean maximum. So velocity is given by minus turn people&#x27;s 40 off velocity to the maximum. We&#x27;ll find the critical points. Doctors will different ship were V and well equipped with you so derivative with respect to be will be minus time Derivative piece one that ever off off 40 0 So we created this 20 So this means that minus 10 is equal to zero. So this is actually an inconsistent equation. What this means is this is actually a continuously in fact, let&#x27;s writer in some other way. It&#x27;s just finder Did it would do so that that ever did this minus strange times. One procedure which is minus stand. This is nothing but be dash. Now, if derivative off any function is less than zero news, doctor tests continuously decrees continuously decreasing. Mies. It does something like this and this is a straight line. So if we&#x27;re talking, if you&#x27;re talking about, it&#x27;s, uh maximum value. It needs that the maximum value it is you did a starting point which is equals zero, which is over here. All can be either are working 10 which is somewhere over here. The negative part indicates start the velocity direction has turned. It is moving down, but we&#x27;re disinterested in the magnitude. So let&#x27;s find out the zero, which is minus 10 10 0 plus 40 which is 40. And it&#x27;s talked about the 10. So we then his minus 10 name stand the S four b, which is minus 100 plus 40 which is minus 60. So obviously this magnitude is more Dando&#x27;s. So when CRT will depend the velocity than strip Indians, velocity has maximum magnitude. Then the party asks us to block the position and velocity drop that respected the time. So let&#x27;s talk about the position Coast Mrs like 60 or 10. Uh, just brutal. Okay, let&#x27;s talk about a velocity first. So velocity is a straight line minus 20 plus 40. It&#x27;s extend a state&#x27;s team deserved Others are people to zero. Its velocity is 14 energy or book 10. Its velocity is minus 60. So this means that it starts from somewhere here, which is 40. It ends somewhere here. That&#x27;s, uh you could ease thes access. This excess course stood here on this point, goes to minus 60. So this goes, something lingers on. This is the point where he is zero. This is minus 10 nt for P and G zero. We already funded in part B zero are people before. So this means that this is be wonderful. So this red one as the velocity All right, let&#x27;s talk about duct Is the position so excusing corporations for the coalition position is minus five piece where just for TT plus under so at equal to zero as zero This will be +00 100 services hunger on as 10 will be minus five times 10 square was 40 times things for under Yes 100. So here we have minus 500 here you have plus five. So the body offices you, Artie on Venice, its maximum a stash is equal to zero as dash is nothing but a visa. When the zero and be a zero from the bar even piece for So the valley office forests minus 44 square less 40 times for us 100 case of this while you come, so does one. Indeed. So now it will be easy for us to locate that. This is for this point. This is maximum. So increased this excess. That&#x27;s his most because then here this is destroyed Trump Green. So this is the point. Greater does 1 50 on a deeper zero. Just hundreds. So let&#x27;s suppose this one is under. So it starts from here. Its maximum care here and then it goes to He was here. So this is how the displacement are. The position and the product evolutionist minus five piece. Where? Less for Katie.

Problem

The total cost $C$ (in dollars) of production can…

Problem 45 Easy Difficulty

Answer

$$80.8 \mathrm{km}$$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Anna Marie V.

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Joseph L.

Integration Review - Intro

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$$80.8 \mathrm{km}$$

Basic Technical Mathematics with Calculus

Anna Marie V.

Caleb E.

Kristen K.

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Integration Review - Intro

Definite vs Indefinite

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Anna Marie V.

Caleb E.

Kristen K.

Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus