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What is the average distance between the parabola $y=30 x(20-x)$ and the $x$ -axis on the interval [0,20]$?$...

Question

What is the average distance between the parabola $y=30 x(20-x)$ and the $x$ -axis on the interval [0,20]$?$

What is the average distance between the parabola $y=30 x(20-x)$ and the $x$ -axis on the interval [0,20]$?$



Answers

What is the maximum vertical distance between the line $ y = x + 2 $ and the parabola $ y = x^2 $ for $ -1 \leqslant x \leqslant 2 $?

We're asked to find the minimum vertical distance between the parade. Bolas Y equals X squared plus one and y equals X minus X squared. Okay, big bar. Well, to do this first nous the vertical distance between these two dfx. This is the absolute value of X squared plus one minus X minus X squared, which is the absolute value of two X squared minus X plus one. You know, it's like now we know that the distance G is well minimized. Black when it's square d squared, which I'll call f is minimized as well. Now the function f of X, this is D squared of X, which is no, it's a piece of shit. I could have two x squared minus X plus one squared Mhm. All right. Now, in order to find the minimum vertical the minimum value of f obey dot com I'm going to take the derivative of F. So we have F Private X is by the chain rule two times two x squared minus X plus one times four x minus one So here is equal to zero. So the critical values satisfy two X squared minus X plus one equals zero or four X minus one equals zero. Now the discriminate of this quadratic equation Delta, this is B Square, which is one minus four times a, which is two times See, So this is one minus eight, which is negative seven, which is less than zero. So this first equation has no real solutions. The second equation has the real solution. X equals 1/4. Now, at this point, we're only considering Nick convinced himself. We haven't restrained our function to an interval, however, noticed that this our function f of X is in fact approaches positive infinity as X approaches plus or minus infinity. Therefore, it follows that F has an absolute minimum on its domain, and it's absolute minimum must occur at one of the critical values. So it follows that F has absolute quite sure minimum at X equals 1/4 and therefore, yeah, the minimum vertical distance is D of 1/4 which is the absolute value, and I'm of two times when fourth squared is 1/16 minus 1/4 plus one. This is the same as 1/8 minus 1/4 plus one. This is 1/8 minus 2/8 plus 8/8 which is nine minus two is 7/8 absolute value of 7/8 which is just 7/8. What? This is our answer.

So Oh, you would find it pointing maker point recurve it. What sequel? Toe vex. That is closer to the point zero. Um, so the picture is, uh, the point. See that you have one here. Two. You have a point down. Good point zero. So you want to find out what is the closest screwed? The licks? What is it? Close this point for these men. Also, you want to many mice the cells. So, uh Well, uh, so arbitrary Point here. He's gonna have some co ordinates x y. So did the sense between ex wife? Yeah. Freeze. You both pieces square these things. That square. Getting these two points would be photo X minus three. So the X coordinate square clothes? Why? According the difference between the White Gordon's, why might zero squared? So because these things, it squirt. Um, so, uh, we find it meaning mobile with the square. He would also be minimum off the function itself so we can minimize that function Me like the square of the distance. That is no problem. It's a world of points along these girls. This point of the form. Thanks, Square it affects because they why she's quote, Vex. So this function of lean well, the the function is gonna beat these. So all of these things is square replacing with the fact that we're on these curves along these curves. Why the sequel to square it affects. So the function that you want to optimize his eggs. Three not squared. Plus quote with picks minus zero squirt. So that is gonna be cool. Deal. Thanks. A square. Minus two times that. So my six eggs waas mine. Well, then squaring this question gives you not eggs for explosive. She's years old of VX, so that defies too excess square, Uh, minus five eggs last night to find the minimum. What you got you here is a word barrel up. So this is those makes us find me here. These are our fees suffered. Agrees, so you can find them. You find out where is his friend? People? Zero. So if you differentiate these, you get two Times X. Um uh, well, if he's gonna do, he's to them. Six minus five. So you said that you go issue if the solution will move there. Five over there. I'll be right till I said that performing that will give you thanks. Should be ableto moving. Defy their surviving last five. You were right to Hi, folks. Oh, so, uh, when the moon, the stars should be 5/2 switches, Probably 2.5. So one. So your leg along there, 2.5. So that is the point. That is the closest from phase here along the skirt. The point by loves on the y Coordinate with me support of that. So why books? So this is the point that minimizes these times on the distance would be, well, air Force that point political this point well being five twelves minds three All this square That distance would be five pumps. Why was three squared close? Square it off. Five Bubs squared, which is gonna be Go 0.5 squirt plus buy a house. So this number is roughly 15 on bad numbers. Oh, this one. He's, uh, 0.25 some that this Times Square would be roughly two point seven flash. So then minimally says the this quarter report that that answer you know, these things would be squared up. Do you want certified

Question 15 we are given an equation of a problem. Why equals? X Squared is the point of zero three and I wanted to find the minimum distance from this parable. Up to this point point would approximately be somewhere in here, and they want us to find the minimum distance to travel. So that's what the quite that's what the problems asking. So when you hear the minimum distance, we need the distance formula, which is our square root. Um, it's minus X of 11 Where? Why? They're all right. So now we're gonna plug in our our point distance equals this clear route. Yes, mine hero. That's why three harmony we're now at this point, I am going to substitute in. We're with you. Why my X squared? I have distance equals square re all. It's weird. Waas X squared minus three. We're and I'm going to feel that and combine all of them. Get this equals this weary of square plus X to the fore minus six X square. Yes, and I combined that meeting it up a little bit and get X fours. Mine's five x squared last nine. Now they want the minimum value for the distance. Um, so I'm gonna take my derivative and with a radical, and, um, I would have to do the change room. However, I know that, um, my equation is with a radical. It does not affect the minimization in the square. Root is always increasing. So because it's not affecting the minimization of the problem, I really don't need the radical sign. So I'm gonna just ignore the radical side and just take my derivative So my derivative gonna be top of the problem would be X to the worth minus 56 square. What's nine equals zero. So my derivative then would become for X Q minus and X equals zero. And I'm looking for my A minimum points so my minimum would be zero or when my, uh, it did not exist, which I don't have any of those points. So I'm factoring out my ex and to have four X squared minus 10 equals zero to either X equals zero or four. X square minus 10 equals zero. And what I saw this. I get it, Hex. Where equals 10 over were, which gives me X equals spearing five over too. Um, now I couldn't reduce that to That's the same thing. I could reduce 10 over 4 to 5 over, too. And then just have that. And of course, that would be clustered minus, um and typically, we're taught not to have the radical sign in the denominator. So that was my instinct just to have the radical in the numerator. However, for this problem, it is easier to work with that five halves under the radical sign. So now what I have to do is find out where it's decreasing and where it's increasing. So what I'm going to do is I'm going to look at intervals. Um, so I have to pick a point from where X is less than this. We're Route five hats. Um, where X is between this were the five have and zero thank you from zero. She's from positive five and flattens were five. So I plugged that in the my graphing calculator. And then I am going to hit my, um, second table set and go to ask. And I'm just gonna ask for the values when X is negative 10 because I just think a random Kuwait that's less than negative square root of five halves, a random point between negative the square root of five halves and zero missing. One negative one. Sorry, one. And here I can say 10 and all I'm really looking for. Is it positive or is it negative? So here, this values and negative this values of positive values and negative in this valley is a positive. And I'm really only concerned when we go from a negative value to a positive value, as that would be my minimum. So my relative minimums would occur at neck. It is 52 and at positive with a y value. Then, of course, that being squared would be five over.

So here were asked to find what is the minimal distance between wise equal to rue decks and the 0.3 halves comma zero. So the first thing we need to do is find what X value is gonna give us the minimum distance. So we're going to start with the distance formula. We're gonna go ahead and plug in the point that we do know. All right, So we plugged in and stuff we know. OK, and I'm gonna go ahead and simplify this a little bit. We're gonna do some distributing simplify and we're left with De is equal to X squared minus two X plus 9/4. All raised the 1/2 power or the square room. All right, so then I'm going to take the derivative of that. Okay? When I take the derivative, I'm gonna have 1/2 times X squared, minus two x plus 9/4 to the negative 1/2 power and then times two X minus two. All right. Was simplified that a little bit, um, we're gonna have to x minus two And the numerator and in the denominator, I'm gonna have two times the square root of X squared minus two X plus 9/4. Now, because I'm finding a minimum, I'm gonna set that derivative equal to zero. Okay, so that's just gonna give me two acts. Minus two is equal to zero, and then X is equal to one. If I plug one back in my formula, I will find the point at which they are. The closest is one comma, one and three halves. Comma zero. Okay, So they asked us what, actually is that distance? So you're going to plug both points into the distance formula, and you're gonna end up with route 5/2 or approximately 1.118 units. All right, so let's look a quick graph to compare these things now, the distance formula. Gonna kind of gonna look like that and then our square root formula and look like that. All right, so if you graphed both of those and you go on your graphing utility and check that point, that point is 1.5 comma, 1.2 to 5, which is exactly rehabs. That is where we were trying to find a minimum at


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