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The Volume of a cube is in direct variation with the side length cubed:If the side length of a cube increases by 17% by what % does the Volume increase?(do NOT put ...

Question

The Volume of a cube is in direct variation with the side length cubed:If the side length of a cube increases by 17% by what % does the Volume increase?(do NOT put a % in your answer)

The Volume of a cube is in direct variation with the side length cubed: If the side length of a cube increases by 17% by what % does the Volume increase? (do NOT put a % in your answer)



Answers

If the lengths of the sides of a cube are tripled, by what factor will the volume change? (FIGURE CANT COPY)

Question 170 here reserving that there is a queue. Yeah. Uh huh. It's the length of deciding piece. Three times I got back to the volume change. Okay so land initially is a paradox. Okay so what it would be so X. You Okay. 11 P. Do we excuse then? Suppose the length become clipper. Okay, so new land as a parent of three in drugs. So new volume Music. Although three XQ. So is it called? Thank you Brexit. Right. 27. Okay. So now by what factor of along jane factor why which Harlem change? There's a party 27 Next to You. My excuse as important. 27 this is the answer. Right? Mm. So you have to find the world even all the factories by the old money conditional extreme by actually we're going to go and decide. What is that? Thanks

We want to write a differential formula that estimates to changing volume of a cube when the edge length change from 10 centimetres, too. Tain 0.5 centimeters that we're going to right first, the formula of the volume of the Cube. And that's eggs to the third, when eggs, cheese, decide the lens of any side of the cube. Ah, that the same thing s the edge blend, which is the same for all the edges of the queue and these volume of the Cube. Then we know that the differential off the volume is equal to the derivative respect to eggs of X Square. I'm sorry for off X to the third time's differential of X, and that's equal to three x square differential of X. So the formula to estimated change in volume of the Cube is differential of be French off volume. He's three times H lands square times differential or variation off a fence. That's the formula. Dad allows us to make estimation a in change involving and now for the giver values. We have that differential. Avi's equal to three times the edge man stand centimeters square times The differential of X is changing edge length and that Sierra Point Cyril, five centimeters. And that's equal to three times 100 centimeter square times. Cyril Boynes, 05 centimeters. And that's equal to 15 centimeters cube And so, UH, T. Estimated changing volume for the Cube. He's 15 centimeter cube when Chikage blends change from 10 centimeters to 10.5 centimeters.

Right now, this probably want to know what happens when the sides of a cube are multiplied by 3/4. We don't know what happens to its volume, so it's looking to cubes here. If we say this 1st 1 has sides of acts, then we get its volume. This execute just our reference here and we have signs of X. The volume is executed. Now take another two. Now, since we want to know what happens when the sides are multiplied by three Force would say this June has sides of three Force x. Yes, we recite. Here's 34 sex. Then the volume would be three. Force s cubed. You multiply three force by itself three times when you take three force to the third power three to the third Power is 27 for the third power 64. So this is 27/64 execute so compared to that original and you multiplied it by 27/64. So, in general here, when you multiply the sides of a cube by three force, its volume is multiplied by 27/64. Just like we saw here

Now. And this problem we have. Well these next few problems are fairly simple. So I'm going to do them all together. They ask us if cuba is expanding with time, how is the rate at which the volume increases to the rate at which the length of of aside increases from the volume of a cube is L. Q. And L. As a function of T. So then V. Is also so we can take V. Dot. And again using the chain rule we get three L squared L dot. So the velocity of the volume, the rate of change of the volume. It depends on the size of the cube. Um you know the length squared and then the rate of change of that length. Now they tell us that we have a rectangle where we have X. Y and Z are the length of the three sides rectangular box. And so we're told that all of these depend on on tea. So we can take basically we can take the that and again using the product rule we get X dot, Y Z plus x Y Z plus X Y Z dot. Now I told that XYZXXYZ. That are all 10 cm per minute. Yeah. So we can basically pull all these dots out And so those are 10 cm to run it. So then we're told what the dimensions are at the time we're looking at because they obviously matter here. And so why is two centimeters E. Is three and X. Is one. So we get all of this here. And this become turns out to be what is the 69, 11. So we get 111 cubic centimeters per minute is the rate of change of the volume at that particular instant because obviously as it's you know as it's changing as these things grow, these things are going to grow. So this is going to change over time also. Um No let's see here we say we have a rectangular plate that's an equilateral triangle. So an equilateral triangle here. And so we have length. I said the sides were length A um Exciting the length inside increases. So a dot is two cm/h. I guess. I didn't write that down here.comals two cm/h. And so they ask is what how does the rate of area increase when the side and A. Is at this instant A. Is 87. So the area of an equilateral triangle is square to 3/4. A squared. Where is the length of one of the sides? We can take the derivative of this with respect to time. And we get Square to 3/4 times to a dot or square to three. Over to a dot. And plugging all that in. We wind up with that. The area is changing at a rate of eight times square. The three centimeters squared per hour again at that instant. And this this is not a constant. This is just at that particular point in time. And finally we have, When the area is square to 75. So what we need to do here is we need we have we need to find we've given the area so we need to find a and so if we do that, well we we invert this thing for a, okay, so you know, we take lined up with the four through the three here square root of A and to hear and then we can plug that into here and we get that capital A dot winds up after some simplifications as the cube group Of, you know, the 4th root of three times the square of a times a day. Now this one ends up being let's see here, we wind up having this is the winds up being the fourth root of 2 55 to 25 times two which is a dot. And that winds up giving us about 7, 7 cm, yeah, centimetres squared per hour. No, this looks this is no, I keep thinking that this is this That's five times 25. Right? Yeah, there's nothing, there's and that's five times five times five it be nice if this was just a huge group but it's not um and I don't think of that. Yeah, it can't be because again, this is centimeters squared. Let's see here, just checking the units. So that would give us centimetres square square root of centimeters. Mhm um this is this is just a constant. Yeah, did I do this right, this is centimeters square. So this this is the right units. Yeah. Did I mess up here somewhere? Oh I did I did mess up, yes. Should be a square root, right? Not affordable. So let's fix that here. I was gonna say something isn't working out with the units here, so that should be a square. Okay, no, let's see here. How can we what is the is there a nice way of doing this? Let's see here. That is we can say that's the fourth root of 75 square, Go to the 4th round of 75 squared. Let's see here Um 75 Squared times three and the 4th root of that four. That doesn't work out nice either. And I was hoping that this thing worked out nicely. But Let's see we have the 4th root of three times the square root of 75. Yeah, And then times two 22.8 and that would be centimeters per hour. Yeah, there's not really uh Yeah, so this should be a square root, I don't know where, how I got 1/4 right over here, So 22.8 cm/h is what we get again. It's always good to check units and that's why I realized something was wrong here because I knew this had centimeters squared, right? And then I was taking the four through to that which gave me where would our centimeters which is kind of weird And I didn't know that wasn't cancelled out anywhere. So this is yet so this winds up being centimetres and this is centimeters per hour. So we have centimeters squared. Obviously. Per hour is the rate of change of the area.


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