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QUESTION ]The follawing dingram which [5 not drawn to scale, represents box of chocolates with dtmensions aS showni The diugrm B, below it shows cylindrical box of ...

Question

QUESTION ]The follawing dingram which [5 not drawn to scale, represents box of chocolates with dtmensions aS showni The diugrm B, below it shows cylindrical box of biscuits with the diameter ofthe Crass- section bcing ZUcmiith un cqual height341 Calculate the area 0f 3 Wrapping paper Ihich can cover the chocalate box: 32 Can the same RTapping be enough for tbe biscuit box in FIG B below?J33 In the following figure; the shaded area is [Orcm?. Find [; the radius of the circle-

QUESTION ] The follawing dingram which [5 not drawn to scale, represents box of chocolates with dtmensions aS showni The diugrm B, below it shows cylindrical box of biscuits with the diameter ofthe Crass- section bcing ZUcmiith un cqual height 341 Calculate the area 0f 3 Wrapping paper Ihich can cover the chocalate box: 32 Can the same RTapping be enough for tbe biscuit box in FIG B below? J 33 In the following figure; the shaded area is [Orcm?. Find [; the radius of the circle-



Answers

Find the area of the shaded figure if the inner radius is 3 and the outer radius is $5 .$ (Such a figure is called an annulus.) b If the inner circle has a radius $r$ and the outer circle has a radius $R$, derive the formula for the area of any annulus. (GRAPH CANT COPY)

Um we are going to start with a semicircle, so we start with a semi circle. Ah, with ah radius of our And what we want to determine is the dimensions of a rectangle that we can inscribe in this semicircle. And the rectangle has maximum area. And this is a church and w and what we want to do. And so we know that area is equal for the right tangle. H times w Okay, what I'm gonna do to make it a little bit easier is I'm actually going to put this in a coordinate plain. So I'm a put this in the coordinate plain where the center of the circle is at the origin. And so I know that now from here to here is 1/2 w on each side. And so I know that this X value right here is actually gonna be equal to 1/2 w and the, um, circle. And so h is, um, is going to be, um, the value when I evaluate the circle equation at 1/2 W. And so the circle equation is X squared plus y squared equal to nine. And so age is actually gonna be that Why Value at the positive square root of nine minus X squared, which is going to be equal to the square root of nine minus 1/4 w squared because X is 1/2 w Okay, so now I can use that appear into my area formula because now H is the square root of nine minus 1/4 w squared and in this is multiplied to a W. So now we want to find maximal area, which means we're gonna have to take the derivative. So the derivative of a the derivative of a Let me get that back into the screen arena for us. If I could do that, I can't do that. So the derivative so a was equal to W times the square root of nine minus 1/4 w squared. So now when I take the derivative of the area with respect to W, of course, is a product rule. So we get the derivative of W, which is one times on the square root of nine miles. 1/4 w squared plus W times a derivative of the square root, which is gonna be 1/2 um, Times nine minus 1/4 W squared to the negative 1/2 times a negative 1/2 W. And so it's kind of clean this up a little bit. So we have this square root, and then over here, this isn't B minus. We have a W squared in the numerator and a four times the square root in the denominator. And then what we're gonna go ahead and do is put it all over a common denominator. Um, and so this would be nine minus 1/4 W square, minus a W squared over four times that square root. And then I also noticed, um, that I can go ahead and distribute that four. So I get 36 minus two w squared over that square root at four times at square root. And so this is my first derivative. And so now what we want to do is fine. Find that critical number associate ID with that derivative equal to zero. Um, and so we get zero equal to 36 minus two w squared, um, and so w is equal to, um three times the square to two plus and minus because, well, it's not gonna be pleasure. Minus is going to be W is a leak, so can't have a negative links. Okay, We also know, um, w has to be between zero and six because my radius is three. And so it can't be any bigger than that. Diameter that circle because it has be inscribed in that circle. And so now if I want to know if this is associated with the minimal or a maximal um, mom and it evaluate minds area zero, which is zero my area formula at three root, too, Which is going to give me a nine and at six, which is also going to give him give me zero. So that tells me that that is associated with a maximum area. And so if w is equal to three route to then H is equal to three over this grow to two, which is three times route to over to, and I just went back, and I used, um h is equal to, um my y value or h is equal to I think it was nine minus. I don't know what I used. Um, yeah, X is equal to 1/2 and then I use yes, h was equal to the square root of nine miles. 1/4 W square. That's how I got that

Nation. States there conceive you, so she's is equal to 54.19 square inches and the height he's given us 4.5 inches. So you know that that so she's area is a little If I r squared less to buy our hitch so it was stolen by you. Do hard square class 20 are two the sequel to Before Going one tonight. So that is our were lush. 20 are equal before 200.1 line Delighted by Do we do? Which is Gillis? 8.6 door. So I asked. Plus 20 r minus 8.62 is equal to zero. Here is what is a C minus? Minus windy? No. So my next 100 when he square it was 400 minus forwarding through all in a 0.6 doing the one that is equal to minus 30 plus 34.4 a. By to their safe will do more in a string t last minus unglued 134.48 I was thinking this will give us celulas 20.8, since for the hideous hostess been is 0.14

We are going to do problem number 11 on this question. A circle is given over there and rectangle is made on the circuit. So the dimensional rectangle is given the given quantities. Ari and Allah is given as five units and he's he's given it as a total units. So what we have to do is we have to find the circumference of the circle. Okay, So as this is the given a rectangle so what will happen is its diagonal diagonal of the rectangle will pass through the center of the city. Let's see, the center is off. So the diagonal of this rectangle will be the diameter of this circle so we can find the value of our C. That is diagonal by distance formula. We can date hyper tennis. That is R C square. That will be equals two R E square, close E c square. So here, R c square that will be close to our is five. So you're right here. Five square, as you see, is 12. So you're right here in the squad. So from here, we're going to get that is 25 close 1 44 that is. Tell a square and under out of this part. So we are going to get on the road off 169. So we got our C. That is 13 units. So here, R c is Demeter. RC is Demeter. So we are going to write the value of its radius. The radius as equals to Demeter by two. That means R C by two. So we're going to have it. Is a 13 by two. No, you got various. You can easily find the circumference of the circle. As we know, Circumference of circle is written as to buy our So we later to buy our our is given to a given to us. We found just right now that is studying by two. So we're going to write two and two bye into 13 by two. So total will get cancel out and we will have 13 by units. Okay, 13 by units. This is the sir components. No, in report, we have to find the area. Okay, Area of circuit. So we have the radius now. So you can right here from here that the expression is by a square. So we will take by a 3.14 and two 13 by two. Holy square, So 13 by two is basically 6.5. So we are going to fight 3.14 into 6.5 in to 6.5. Now we will calculate this value. If we calculate this value, we are going to get 1 32 point 665 United Square. You cannot help like this unit is sick or whatever it is, you know, it would be We have read that you need and square. You have to put a square now. It is certainly a question too, right till its 10th nearest 10th. So we are going to be just right it till nearly 10. So that will be 1 32.7 Unity Square. So this will be the answer. This will be the area. Thank you very much.

In this kitchen figure is given the side of square Which is seven cm. And billy is an arc of circle of radius A. B. So here you can see that it is a circle the true radius A. B. So we know that length of baby which is seven centimetre because all side of square are equal. So after that in the execution we have to find the area of shaded region. So first of all we find the area of sector A. B. D. So here area all sector A. B. D. Equals here. You can see that two days, 90° so 90 degree upon 360° times. Fire is choir. So here we get one upon 4 times. Fire is choir. That is the area of sector Evd. You can put the value of our which is 74 times by Times 7 to the power to. And here you can see that we have a right triangle which is A. B. B. So we find the area of sector sorry, triangle E. V. D. Equal to 1.2 times base which is seven. And hide which is seven. So now you can write like that 49 Upon to. So now we can get negotiated region when we subtract area of sector Abdi from area of a triangle A. B. D. So here we get yellow shaded region. So you can see that both regions are equal. So We multiply by two. So now required area equals two times area of sector A. B. D. Negative area all triangle A. B. D. And now we put the value of this. So here we get two times one upon 4 times 22.7 times 7 to the power to negative 49.2. So now we simplify this. And here we get two times 28 upon two equal to 28. Same esquire. So you can see that the area of shared the region Which is 28 same square which is over Wild answer option three


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