G2 Topic 2-5 Effects of Changing Dimensions on Volume
ok, we are in section 25, effect of
volume, and let's find the
volume of this prison. Remember that the
volume formula is the base area times the height and since it's a rectangle the base is the join times the width So I'm actually going to change the capital B to LW because that's going to be your base area, so just make the connection here times width times height so it will be 5 times 2 times 4 10 and 10 times 4 is 40 so 40 cubic feet would be the
volume of this prison. Okay, let's look at what will happen if we change one or both
Let's go ahead and write 40 for our original and what will happen then is us. We're going to change a few things. We're going to make a link that's twice as long, a width that's twice as long, and a height that's three times as long. Okay what will happen this time when the length and width and height change by twice as long for the link that will make The link 10, the width will be twice as long so four and the walk will be twice as long , unfortunately three times as long, so it's going to be 12, so 40 times 12, so the
volume of the new prison would be 480 cubic feet.
Let's go ahead and end the chart in this column here. You want to change the link 3 times as long as there is no change here so let's just say x 1 for the width and twice as long for the height so your length times your width times your height on this here your link three times more than five that 15 is no change in width so leave a 2 and your height is twice as long so that's going to be 8 will be 15 by 2 that's 30 by eight that would be a
volume of 240 cubic feet. Let's talk about the next one. the next thing they're going to do, the length is going to change four times, we're going to multiply half the desire and we're going to multiply the height three times, so on this here your new
volume 4 times 5 your length is 20 which would be half your width 1 and your travel is three times larger, that would be 12, so 20 x 12, so your
volume comes out to 240 cubic feet, ok we're going to notice a pattern here, we've done this with area before, but let's take a look this on.
volume is too high. I'm sorry for 80 if we divide that by our original 480/4 you get 12 on this your new is 240 your original was 40 and 2 40/40 is 6 on the last one changed it's 40 /40, that can also be six, okay, I want you to notice what's going to happen here, you take these
dimensions and change them by 2 2 & 3 If you multiply this two times two times three, notice that you get 12 here three times one to notice you get six and again four times a half times three which will give you six so that will help you when you notice the guess here would be that that new
volume multiplies the original floor by the product of the factors.
On the next page, write what you can guess about the effect of
dimensions on the
volume, and let's write that down. Okay, the
volume. To get the new
volume, multiply the original
volume by the product of the factors by which you change the
dimensions and we noticed that in the type of paid spreadsheet we just made. Let's try assuming the
volume of a right triangular prism is 360 cubic units, what would its new
volume be if one of its
dimensions was twice as long, a second dimension was three times as long, and a third dimension was half as long.
Note that they give you the different changes. Let's write down our changes then give half the time to think that it is the product of these factors that you are
changing with. So if you multiply six times half you get three so take what your new
volume would be what the old
volume was the original
volume multiplied by the product of the changes giving our new
volume which would be 1080 and cubic units , so really just look at the change in all
dimensions and multiply that to find your factor that will make your new year
volume Let's look at example 3.
volume of a cube is 54 times the square root of three cubic centimetres, what would its new
volume be if one of the
dimensions was halved, the second dimension doubled and a third dimension was not changed, so below we write the changes again on this the new one
volume has one of its
dimensions the second has been doubled and the third has not changed now if it doesn't change remember to put the factor of one if it doesn't change so half times 2 is 1 and 1 times 1 is 1 note this each times you get a 1 which means there is no change because if you multiply by 1 you stay the same so your new
volume will be the same as your original it will be 54 times the square of the square root of the Triple your one, which simply means it will remain 54 times the square root of three.
I think I'll give you the decimal as well. Let's put that into our calculator and see what that would be. That's an accurate answer and our estimate is about 93.5 and these are cubic centimeters so we can write it as centimeters to the power of three and that's how you determine the change in
volume when you change something. so you just take the product of the factors about what you're
dimensions, okay, now it's time to do some work on yours