The Position of a Particle Is Given in Cm by X = (2) Cos 5ã€t, Where T Is in Seconds
Question
(a) The position function of particle is given by s(t) = & 9t2 + 15t + where is measured in meters and in seconds_ At what time(s) is the particle at rest (i.e. velocity is 0)?The radius circle is increasing rate of 3 centimeters (cm) per minute_ Find the rate of change of the area with respect t0 time when 6 cm . Leave your answer in terms of pi and include units
(a) The position function of particle is given by s(t) = & 9t2 + 15t + where is measured in meters and in seconds_ At what time(s) is the particle at rest (i.e. velocity is 0)? The radius circle is increasing rate of 3 centimeters (cm) per minute_ Find the rate of change of the area with respect t0 time when 6 cm . Leave your answer in terms of pi and include units
Answers
Assume that a particle's position on the $x$ -axis is given by
$x=3 \cos t+4 \sin t$
where $x$ is measured in feet and $t$ is measured in seconds.
a. Find the particle's position when $$t=0, t=\pi / 2,$ and $t=\pi$$
b. Find the particle's velocity when $$t=0, t=\pi / 2,$ and $t=\pi$$
Well, hurry, incompetent can be calculated. So, really, the radial component is equal to our double. Don't minus our Peter Don't screen and tangential component is Peter. Well, anti dies equal to our creditable Don't less to our don't Peter, don't. We can get ti the daughter anti datable dog. So Peter don't is equal to trade because sign of three t and Peter double don't is able to minus nine off sign three t well, and Tina is equal to 30 degrees. We have o t is equal to signing worse or 30 degrees divided by three, which is equal to 10.5 to 5 on second uh, now Peter, the patina dot is equal to 2.5559 per second. And creditable don't is equal to minus 4.7 12 for our second square right, And further we calculate the ready lex relation, so ready lacks relation is equal to zero minus six, multiplied by 2.5559 square, which is equal to minus 39 going to 196 and a tita is equal to minus six, multiplied by 4.7 1 to 4 places Ego which is a good to minus 28.274 and talking X relation is given by the root. A squared are This is Square Peter, which is equal to root. 39.196 square less 28 point 274 square. So daughter Lex Relation A is equal to 48.3 inch or seconds.
Hello, New radio. Welcome back. Okay, so we're in chapter three, Section nine, pitch to 49. This is question, um 22. Oh, Stewart's capitals with early transcendental seventh edition. So have a particle moving along this curve. Okay, Now, at the moment that the particles move passes through the 0.1 3rd comma one, let's say that the X coordinate the X DT is moving at a rate or changing at a rate of red tents, and we just for a second. The question is, how fast is this particle moving from the origin? Well, let's say that the distance capital D that the point is from the origin is X squared plus y squared, isn't it? Already? So we wanted to do is figure out the right to change of d so d cap D d t. I'm sorry. Square River, which is X squared plus y squared to the one half, right. Oh, boy, this is missing you. So it's it's differentiate that implicitly respected time. Using the chain roll stuff to the one half gives you one half stuff to the negative. One half tend to do with this stuff, which is gonna be two x x prime I write it dx DT plus two y my primary duty. So to figure out the the kept d d t we need to know exits one third. We need to know why it's one third two times like this Two times one third dx DT is red 10 2 times Weighs two times one and we need to win it. Er, we don't have to read it. Okay, so let's get the GTO. Since why is to sign of stuff de y t t it's gonna be during the sign of stuff is through the constant times, the functions that constant times that during the functions of two times during the Sinus co sign during the sign of stuff is co sign of stuff times to do with the stuff inside. Expect tax is gonna be five to the two is canceled, right, the two in front of the coastline. So I get hi terms of co sign of, uh, pi X over to now. Since X is a third since X is a third at the 0.1 3rd come of one, we get pie times the cosine of pirates six. I'll wait. Don't forget the river. The inside includes ex prime. So this plug in X is one third five with two times 1 35 or six turns red 10. So you have, um hi. Ride 3/2 times. Read 10. I guess you could say that's read 30/2 times. Apply. That's a 30. 0, all right. So now D cap T D DT at the point. One third come a one on the curve y equals, Um, what was it to sign pilots over to is going to give you one half as we have up there. One half of X script was white script. So one half of one third squared, which is 1/9 plus one squared, which is one times two times X, which students of third is two thirds times the ex city, which is rad. 10 centuries for second, plus two times. Why which two times one is to times divided. D, which is rad. 30 over to hi to the twos. Cancel and that's a huge mess. Well, let's see. Hang on. Did something weird. That's one half X squared plus y squared to the negative. One half, right. So there's a two here. And there's a negative one half here. Uh huh. So what is that? Well, 1/9 plus 9 +96 10 nights. And so the negative one half would be 9/10. The skirt of 9/10 would be three over. Red 10. Viggo, these two is canceled. This is just a mess. All right, So this is just to read 10/3 plus red 30 pie. I don't think we could do so much more. The play is not in the radical. And if you bunch of trust a catheter, it's about 9.2 centimeters teas and semi right for second teas and seconds. Yeah. Okay, so that was helpful them. Right? Have a nice day. Good luck with your studies. Bye. Bye. See you next time.
Nice of the position of a particle on the X axis is given by X equals three. Curse NT. That's for Sye Inti. We want to find the particle's position Civil, right, The sex of team. So you want to know x zero ex of pie over too and exit by. No, just play him in x zero is three that zero survivor to his floor. Texas papayas minus three. Okay, And then be we want to know the particle's velocity. What's the derivatives? So let's take the derivative ISS Negative three sign T spore Curse nt and so x prime of zero is for six. Prime of pie over to his mind's three Next private pie is what for? And these air always in Okay said position is in feet Kong X axis and the velocity is in feet per second.
Were given a displacement function. We're told that X is three who signed T plus four Cy Inti, and I'm just gonna write Ex Prime, then is minus three Cy Inti plus four co scientist because we're asked to do something with both of those. So were asked to find both of them when time is equal to zero. So X is going to be three times the co sign of zero plus four times a sign is Euro. And if you know your special angles, which hopefully you do you know that CO side is one at that point, and sign is zero. So we just end up with three, okay, and it's a unit of feet per second. If we find Ex Prime, which is the velocity which were asked to find that it's minus three. Sign of zero plus four co sign a zero, and the sign of zero again is zero, and the co sign of zero is again one. So we just end it with four times one, and that's four, and that's in feet per second. I messed up this unit. It's not in feet per second. It just in feet. It's a displacement a member has to find the same things. But at T equals pile or two. So again, our displacement There's gonna be three co sign of pi over two plus four. Sign of pi over two And the coastline of pi over two is zero and the sign of power to is one. So we end up with four feet as our displacement and our ex prime, which is Air Velocity, is again minus three. Sign of by over two and then plus four co sign of pi over two. And since we already know what those are terms of signing coastline if I were to we know what his minus three feet per second because that is one and that is zero. And then finally we're asked to do when t is pie. So what did the same thing again? Exit is going to be three co sign of pie must fort sign if I if you then evaluate the boats on the I minus one. And the sign of pie is zero. And so, Anna, with minus three feet as my displacement in my ex prime which is my velocity is minus three times a sign by plus four. I was the coastline. If I and we know what we just did that zero and this is one and we end up with mine. It's for feet per second, and that's the answer to the whole question.
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The Position of a Particle Is Given in Cm by X = (2) Cos 5ã€t, Where T Is in Seconds
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