Wenk Geometry Lesson 11-7 Effects of Changing Measures in Different Dimensions
So the goal for today is to describe the effect of dimensional changes in
dimensions. We've already seen how elements in the first dimension change the width or height in the first dimension, what effect this can have in the second dimension. Now let's look at what effect it can have in the third dimension. I think we've already seen that when the measurements include height, length, width, depth, perimeter, although it goes around an object and looks like it's a 2D actually
measures a 1D measure, so a perimeter of circles . These are all one-dimensional measurements. They are measured in feet, inches, miles.
Then he has to take
measures which is a great form of area which is measured in square units finally 3D measurements are measured in cubic units cubic inches cubic centimeters cubic miles and you are talking about volume or capacity so when you are usually looking at a scale factor when you are using the Scale say ng factor as in this case are scale factors because everything doubled the radius than the height you usually gave this scale factor in one dimension here this triangular sorry square pyramid what is the scale factor here? out of five these are one-dimensional scaling factors. similar solids are the same shape but not necessarily the same size, not all cylinders will be similar as you could have a short thick blunt cylinder or a really tall thin cylinder with pyramids, but some figures will always be similar, like for example a ball.
All balls will be similar. The scaling factor here is seven if you go from small to large. If you go from big to small the scale factor would be one set here we go from big to small so what is the scale factor it will be 3/5 and all the cubes will also be similar so here we have two Questions, obviously we said similar cylinders are not all similar so the silk cylinder here will be a B or C similar to the first which has a six to ten ratio but it reduces to three to five. So if you compare the ratio to height what other cylinder would have the same ratio of three to five obviously not a but B is clearly three to five link to wid only in order from smallest to largest two to three to five let let's see how a ranks let's see if I reduce that okay and if you reduce I don't think it can be reduced so that's definitely out how about the twelve sixteen that can be reduced by 4 2 2 3 2 4 that's a lot nah that's almost similar that's what's similar because it reduces by five you get the same ratio so now let's see what happens when you jump from one dimension to another so the scale factor here and 1d will be two but if you go to two
dimensions you know you are comparing the footprint or surface area that the scale factor is going to be two squared it will be four times that if it's not just in has grown in length but also in width, so it has grown in two directions and with 3d that scale factor will be two cubic centimetres, if we go on to a sphere which in 1d increases by a scale factor of three.
What effect does this have on a 2d measure such as surface area it will be three square to volume it will be three cubic this anxiety increases by a scale factor of five in one dimension. So if you compare their circumference or know if you knew the 5x increase in radius the effect on the circumference would be a scale factor of five but the 2d measure let's say the surface area or the area of the great circle which is a circle which is the sphere in half which would be five squares or 25 and then the volume would be five cubic meters so basically the rule is this if the length of a solid increases by a scale factor K then the surface area increases by K squared and the volume um K to the cubic centimeter so that's a pretty important fact and I'll give you a moment to write it down if you need it but basically the essence of the whole
lesson so let's go to see which ones
Effects of increasing certain
dimensions of a figure on elements with
So if we increase the radius by a factor of four, if the radius increases four times, what happens to the diameter of a sphere? Well if the diameter is one dimensional so it will be four times larger what will happen with the surface of the sphere so that is a 2D measure so it will be four squares or 16 times larger what will happen with the volume four cubic or be sixty four times larger perimeter what happens is the comfort of the sphere good this is a 1d measure so it will only be four times larger so the length width and height of a prism will be tripled which equals h to the base perimeter perimeter that is a 1D measure so it just gets tripled what happens the surface area which is a 2D measure so it gets three squares or 9 times bigger and the volume is a 3D measure so it gets rolled three or twenty seven pounds bigger okay i'm trying to give you a better picture of whats actually happening here so i have a sphere this is my original sphere has a radius of ours what happens if i double the radius well, the radius and i i scha I'll see the effect on the volume, so the radius is doubled, but if you think of a formula that rolls that radius, 2r is basically multiplied by itself three times, so you'll see that it actually adds up to eight Times bigger and what happens if I triple the radius well back you triple it in three directions, three directions that affect the volume so three cubes are divided into three and that's why it's 27 times the volume, okay, so this is a classic problem that will be on the test and there are some of these in the homework as well.
They give you a one-dimensional scaling factor. They say that the figure will change as a result, and then they ask what effect that will have on the surface area, which is 2d, or volume, which is 3d, so here they say that the rectangular prism will change by a scale factor of 1/4 is shrinking so now it's getting smaller which is the 3D effect of the same scale factor you're going to have to cube the scale factor so it's going to be 1/4 cubed so the volume will basically be 1/64 of that what it was so my original was the big one it was 200 cc so to find the volume of the smaller one I compare small to large and come up with a simple proportion that I know my 3D scale factor is 1 in 64 is and that would equal x over 200 which has to go down 200 because that the volume is a larger number i just cross multiply and solve and we get our answer 3,125 cc now you should pause the video here and ver search it with this one so the scaling factor will be 3 bigger.
It doesn't really say the problem is a 1d scale factor, but if it just says the scale factor is 3 you can assume it's 1d unless it says otherwise the would be 2D -Scale factor we're looking at here on the surface, 9, that would be 3 squared, so the small again was 40, that's the small, again, I'm comparing big small to big, and my scale factor is 9, so the new 9 -times bigger my ratio and in 2d will be 1 to 9 since 40 is the ax. Forties have to go to the top, that was the surface of the little one and I cross multiply and solve and we see that the new one of the bigger ones will have a surface of 360.
Okay, the trickiest problems of all are these in my opinion. They give you the ratio in 2D. They indicate surface conditions. This is a 2D measurement and then they ask for the ratio of their volumes now maybe you can go from 2d to 3d in your head but i can't so i'm gone. Well jump back to 1d when my 2d ratio is one to nine then my dear it has to be the square root of what 403 would be and now I can figure out what my 3d ratio is just by rolling one two three and I get a 227 now if you in your head can go from 2d to 3d, good for you.
Let's do number two to descale the factor 9 to 25. This would mean that the scale factor of 1d would be the square root of three through five and a 3d scale factor would be the cube of 3 through 5 27 through 125 number three through DS four through nine Wendy would have to be two through three and then three D would be tequila, which looks like it's 8 to 27 number four pretty much the same as me I think you get the idea here that that's a big one and ultimately your mistake of the day. i have a good one Our test is coming up and I hope you're ready for it.
I think I think you'll be fine