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Question

In the following exercises, solve the given maximum and minimum problems.
Computer simulation shows that the drag $F$ (in $\mathrm{N}$ ) on a certain airplane is $F=0.00500 v^{2}+3.00 	imes 10^{8} / v^{2},$ where $v$ is the velocity (in $\mathrm{km} / \mathrm{h}$ ) of the plane. For what velocity is the drag the least?

Ernest Castorena · Accepted Answer

Okay, This family, they say that we wanted to calculate minimum run railings or an airplane with using airplanes acceleration and take off speed. Eso There&#x27;s going to be a three step program for when things first. ISS. Let&#x27;s see from user for input. Yes, we&#x27;ll use the consul is counsel and we&#x27;re going through never toe convert steering input and the numbers to do calculations. And then we&#x27;re going to calculate we minimum run railings We&#x27;re gonna use the formula. They&#x27;ve given us length for the equal to, Let&#x27;s see, be take off speed So right, speed squared Then we&#x27;re going to you divided by yeah times the acceleration And then the third step we&#x27;re going to display output Teoh user This case we&#x27;re going to use Consul And remember Teoh convert two strings but in the opposite of what we did taken inputs right? Let&#x27;s see, they give us an example when we prompt the user we&#x27;re going to need to get pliancy speed on acceleration We&#x27;re going to even see to turn our string input into numbers that we can calculate with we&#x27;re going to ask for in print Let&#x27;s see asunder speed and acceleration and it looks like the user is just gonna have to know that they should call on separate there. Okay, so we&#x27;ve run that at that point, and I were going to calculate it. Uh, I have to say that links going to be equal to, well, im formula taking it. It was like a power under going to divide 0.2 times the acceleration that we got. How are we going to display that to the using? Well, from their output, we congratulate into the conflict. Bring us A. The minimum runway link for this airplane is and they&#x27;re going to give the number. They calculated our length, but remember when he had turned it into a string here? Right. So who knows how to combine a bunch of strings Or else it wouldn&#x27;t know how to combine our string message on the number very easily. Then let&#x27;s see you give that number and they make sure they let them know that they converted it. This they&#x27;re probably didn&#x27;t even ask the user and let them know the units. That&#x27;s a bit interesting. You should always let them know. So let&#x27;s actually improve on what they showed us. Let&#x27;s say we want in the speed and me. There&#x27;s for a second. And let&#x27;s say that we want the acceleration in meters per second squared. Yeah, violence maps their input and see if we get the same out. But when they put it in, they put in for their test 60. Um, I agree with the comment three point by over the acceleration. So it&#x27;s in that, and they end up with that. See, the minimum run away for the airplane is 5.1, important to eight and Aaron six. Okay.

Oswaldo Jiménez · Answer

we will find the absolute minimum and absolute maximum values as a function F of T equal to co sign of people. A sign of two times T on the closed the interval from syria to my health. Without this function is continuous in this close interval. And for that reason we know F attains its extreme values at points on that interval. In fact we know those points were F. Attendance. Extreme values may be either the end points of the interval or critical numbers of the function. For that reason we need to calculate the critical numbers of F. For that we calculate first the first order derivative of F. And that is negative to sign of T plus two co sign of duty. And now deserve a tiv is defined everywhere in the real number. So F derivative of T exists four every T in the interval zero by health. And for that reason we can say that The only critical numbers of f are those values of T. four which The first derivative is equal to zero. Yeah. So that reason we get to solve this equation for preservative of F equals zero. And this is equivalent to the equation negative to sign of T. Plus to co sign of duty Equals zero. Which is the same as um go sign of duty equal. Or in fact we can say instead of that All 10 of TTINA son of T Equal to zero. And now we can use the formula of the co sign of the double of an angle which we know is equal to Coulson square of t minus Saint square of team. Okay And that minus sign of T equals zero. and now cause square of tea is one minus sine square of tea. That minus Since we&#x27;re t minus sine of T equals zero. And then this is equivalent to the equation 1 -2. Sine Square of T minus sine of T equals Iraq. Which can be written as to sign square of T plus sign of team -1 Equals Syrup. And now we can make a change of viable here we named alpha good sign of T. Or maybe they let&#x27;s not use alphabet and other letters that matter like em Then we get to um square plus M -1 equals zero. And we can solve this equation using the formula we know for the solutions of this equation which is equal to 91, more or less square root of one square which is one minus four times two times negative 1/2 times two that he was negative one more or less were off one plus 98 over four That give us 91 more or less periods of nine. Okay skirts at nine Over four, that is 91. More or less. 3/4. So we get em one is nearly one plus 3/4 which is three minus one is two of the four is one half An M two is 91 month for over four. And that is -3. I meant sorry here. Mhm minus three And that is negative for over four is -1. So remember a sign of tea. So we get sine of T equals one half. Or sign of T Equal 91 4. T. In the interval. Remember the domain 0x2? Okay so we got to remember the sine function a little bit here. Over the interval syrup by half we have Sign of Serious zero. So you get increases let me throw this smaller here. So let&#x27;s say get this by half. We know sign is equal to one and at 00. So here we can say the value is one. So this is sign of T for this interview. And at that interval this solution has no equation has no solution to this equation has no solution because we have sign a loose positive or zero. So this cannot be and so sign of T equal one. Have you see angle for which we have This value one else here and we know sign of um by six is one half. And The reason we can say that tea is 56 and it&#x27;s the only solution. In fact because graphically we can see that there are no the points in the interval zero from zero by half where a sign of tea is one half. In fact functions include increasing. So it&#x27;s 1 to 1 or there. So there&#x27;s only one value of T between Syrian health for which the image is going to one half And that number is 5, 6 belonging to this interval syrup by half. We have this critical number and that&#x27;s the only critical number as we said here both the only solution in the given interval of the equation for the difficult zero is the only critical number of F. So the only critical number of F Ian zero by Health S. T. Equals 5 6. So we got to evaluate the function at that critical number and at the end points of the interval let&#x27;s start by evaluating F0 and my health efforts zero is to go sign of zero plus sign of two times 0. And that is because in a serious one we get to send a serious serious now at the writing point by by health we get to go sign up by half blast sign of two times by half. So this value here is equal to zero and here this value is equal to one. So we get to let&#x27;s see because I&#x27;m serious. One authority I made a mistake here Try to say that this is zero here But this is zero also here because co sign and zero is 1 and then by half as we see here zero. So we got zero here&#x27;s here here so we get zero sorry for that. Okay. Yeah the image or the images of the endpoints and now we find the image of The curriculum number by six. And we get to go sign of by six blood sign of two times by six. That&#x27;s the same to co sign up by six plus sign of five third. That is to square 2 3/2 plus square to three over to. And that is Once for 2 3 plus scores of 3/2 is three squirts of 3/2. And that&#x27;s about the value here. It&#x27;s about 2.59 8076 21135 3316. Okay so we have this value here and these are all the values we get to consider. So the maximum the largest of the values, the absolute maximum value of the function over a given interval and that is 2.598 etc. Which is the the image at the critical number 56. And the minimum the absolute minimum value of the function which is the smallest of the three values Is zero attained at the writing point by half. So we write that answer here then. D uh absolute maximum that&#x27;s right. First the minimum to the minimum value of f In the interval zero by half is zero. Which of course okay. At the writing point as we saw here, yeah at the right in point t equal by uh my health and the absolute maximum value of the function in the interval zero by half is three scored 3/2 which is about Which is about 2.59807621135 3316. Which of course diverse and By six which is sick. The critical number also function T equal 56. So exist the answer to the problem. And we remember then procedure here was to verify first that the functions continues over close the interval. And for that reason we know we detains its extreme values on that interval and knowing that the extreme values may be attained either at the end points of the interval or at critical numbers of the function. We start by calculating the first derivative of F. And then we identified the preservative exists for everybody of team the given interval. And so the only critical numbers of those parts of the argument. T. For which the functions first toward the derivative of physical zero, stating that equation for sort of derivative of F. At any T equals zero. We solve that equation and we got two possible solutions to equal uh The equation is equivalent to sign up to equal to 1/2 and sign up to equal nearly one. The equation center Chico native one is impossible for tea on the interval syrup by half because over there The functions sine of T. is always positive or zero. So the question we have to solve finally is sign of equal 1/2. And the only solution on the intervals by half of that equation is 56. So the only vehicle number of f in the intervals by half by six. And now we get Then to the function at the end zero I have And at a critical number by six we did that we found that the largest value of the function on those three points is 2.59, et cetera. With That is three score to three over to just the image of the critical number 56. So that&#x27;s the absolute maximum value of the function over the given interval and the smallest of the three values is zero. So it&#x27;s the absolute minimum of the function over the interval. And that of course at the writing point of the intervals uh syrup I have which is by half. And that&#x27;s the end the solution to the given problem.

Sajin Shajee · Answer

all right. We&#x27;re close to wife. Where means why? Thanks. Why? G of x g x Postwar minus Using this, you&#x27;re solving the reason the government the minimum. So for greedy that we can call to you eggs. That&#x27;s why I four y she rating that she could eat. I chair you x me is why he cooperate for why these x equals five. Why cold three x My three thanks is equal. Okay, so this will be. But during a yearly, he extralegal survive. Why three fine? Three equal to 20 years which is shown as Tell me why you&#x27;re keen. It is to a point.

Mike Gaerlan · Answer

Okay, so we need to find the maximum volume where the volume of a suitcase is gonna be. Link times with times height, right. We&#x27;re also restricted by the fact that length plus with plus height, is equal to 62. So we can rewrite one of, um either length or with in terms of the other because hiatus fix. So we&#x27;ll just say length is equal to 62 minus w my insulin times the height. So we get the volume is equal to will replace length with what we just found. Some 62 minus with times, height, um, times W times h So simplifying that a bit we get 62 w h minus w squared H minus. W h squared. Now, uh, here are with is our variable. So we&#x27;re going to take the derivative with respect to whip so d v d w. Here. So taking the derivative with respect to W, we had 62 h minus two w h minus h squared. Remember, we&#x27;re trying to optimize, so we need to set the derivative equal to zero, and we&#x27;re solving for H. So we get to w H is equal to H squared minus. Sorry to W H is equal to 62 h minus H square. So divide by two h on both sides to aged about by two age here we got that W is equal to 31 minus 1/2 a tch. Okay, so then to find the link we plug in, huh? You plug in the whip here. Okay, so we get 62 minus 31 minus 1/2 each, minus h. Okay, so we get 62 minus 31. It&#x27;s going to be 31 and then this is plus 1/2 a tch within minus each. So we&#x27;re left with 31 minus 1/2 a tch. So that&#x27;s the same as W so Alice equal to W in this case. Now, to find the volume we plug in w N um, W l So, since lengthened with are the same, it&#x27;s 31 minus 1/2 a TSH squared times each. So this is our maximum volume, given a fixed hype. Now, if we want to maximize this with respect to height, so we need to find the height that maximizes this. Okay, so let&#x27;s go to and you white board here. So We know that, um, volume is going to be equal to 31 minus 1/2 each, squared h. And now we want to optimize this respect a height. So we need to find the derivative of this with respect. A height. So this is going to be we&#x27;ll use product rule here. Um, so first it&#x27;s gonna be the, uh So the first times the derivative the second, which is one plus second times every one of the first. So two times 31 minus 1/2 a tch. Times negative. 1/2. Okay, simplifying this a bit. We have 31 minus 1/2 a TSH squared, plus our Sorry, we can, uh, minus. So the two and the 1/2 cancel out. So we have left H times 31 minus 1/2. It&#x27;s okay, so Ah, we can set this equal to zero here. Um, I hope there&#x27;s a type of here that should be 31 h squared equals zero. So said that equal to zero we can factor out a 31 minus h squared. Um, so we factor out 31 minus Well, 1/2 hh. Okay, so we&#x27;re still not for 31 minus 1/2 age here. If a 31 um minus 1/2 a tch here and then minus each is equal to zero, so we can set either these to be equal to zero. So this is 31 minus 1/2 H is equal to zero, or here we have 31 uh, minus minus three halves. H is equal to zero. Okay, so here on the left side, first we get that H is going to be equal to 62. Okay, it&#x27;s for the for the next one, right? We have three halves. H is equal to 31 or three h equals 62 or we get H is equal to 62 divided by three. Now, if we have a TSH is equal to 62. Um, for this solution here, um, if we plug in or if we try to plug in for W and H, or if we try to find the volume here. So here volume is equal to 31 minus 1/2 of 62 squared times 62. This, um, this part is 31. So we get 31 minus 31. Uh, so it&#x27;s just zero. This is just all zero. So the volume zero. So that&#x27;s not what we want. Um, but if we plug in each over 32 instead h over 62 over three. This will give us our massive in volume. Okay, so the dimensions are, um Well, this is the height that will give us the maximum volume and not, um, h equals 62 here. Okay. Um, okay. Um right. Not this one. Okay, so there we go. We&#x27;re done.

Daniel Nolan · Answer

Okay, so I question here is actually was to find the maximum or minimum values for this function using the maximum minimum values formula. Okay, so we&#x27;re first going to find the X value of the Vertex. And then we saw that in tow, Our function to find our maximum or minimum. So using the equation and the for ministry of, uh, minus B over to a we get the X equals minus 80 over minus two times by minus 20. I saw a positive two times by minus 20. Okay. And then that equals minus 80 over minus 40. Which gets us to so then we get X equals twofer are Vertex on Deacon Silver into our equation to get our maximum or minimum value to get minus three plus 80 times by two minus 20 times by two squared. So that gives us minus three plus, um, 160 minus 20 times by force of my 80. So we get, um, 1 60 minutes, eight, which is 80 minus three. So let&#x27;s go to 77 for a maximum on minimum value. And since that Graff is a on negative, uh, s so it would look something along the lines off this&#x27;ll there were actually a maximum value. Here is our maximum value is 77. If you have any questions about that, you can email me Daniel No J. P H doc Nolan gmail dot com I wish you the best for your studies. Have a great day.

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In the following exercises, solve the given maxim…

Problem 42 Easy Difficulty

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

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus