Aug 23, 2022

# Wenk Geometry Lesson 11-7 Effects of Changing Measures in Different Dimensions

so today's goal is to describe the effect of

## changing

the measurements in

## dimensions

, we've already seen how elements in the first dimension change the width or height in the first dimension, the effect that can have in the second dimension now We are going to see what effect it can have in the third dimension. I think we have already seen this: when the measurements include height, length, width, depth, perimeter, even though it goes around an object and it seems that it would be a 2d, it really

### measures

a 1d measurement, so a circumference of circles these are all one dimensional measurements they are measured in feet inches miles then he had to make measurements which are an amazing form of area measured in square units square mile square inches could be base area surface area and then, 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 look at a fact or of scale normally when they say the scaling factor as in this case they are scale factors because everything doubled twice the radius than twice the height, usually that scale factor is given in one dimension here, it's triangular, sorry , square pyramid, what's the scale factor here, is it five, everything increased by a factor? out of five these are these are similar solid one dimensional scale factors will have the same shape but not necessarily the same size not all cylinders are going to be similar because you could have a short fat stubby cylinder or a very tall thin cylinder the same with pyramids Without However, some shapes will always be similar, such as a sphere.
All spheres will be similar. The scale factor here is seven. If you go from small to big. If you go from big to small, the scale factor would be one. here we go from big to small so what is the scale factor is it going to be 3/5 and all the cubes are also going to be similar so here we have two questions here obviously we said similar the cylinders are not all similar so which silk cylinder here, a B or C, it will be similar to the first one which has a ratio of six out of ten, but it is reduced to three to five, so if you are comparing the ratio to the height, what other cylinder would have the same ratio of three? to five, obviously not a, but B clearly is from three to five C, what does that simplify for you?
Now let's look at the rectangular prism that has a four to six to ten if you compare the height to the width link in order from smallest to largest from two to three to five let's see how it ranks let's see if I narrow that down, okay, and if it reduces, 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 is very close that is almost similar this is the one that is similar because it is reduced by five you get the same ratio so now let's see what happens when you jump from one dimension to another so that the scale factor here and 1d is going to be two, but when I go to two

## dimensions

, you know that you compare the base area or the surface area, that the scale factor is going to be two squared, it's going to be four times larger, not just it grew in length but also in width so it grew in two directions and with 3d this scale factor is going to be two cubed if we go to a sphere that increases by 1d by a scale factor of three what effect will that have on a measure of 2d such as the area of ​​the surface will be three? squared the volume it will be three cubed this fear increases by a scale factor of five in one dimension so if you compare your circumference or you know the radius increases five times the effect on circumference would be a factor of scale of five but the 2d measure say the surface area or the area of ​​the great circle which is a circle that cuts the sphere in half would be five squared or 25 and then the volume would be five cubed 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 by K cubed so that's a pretty important fact and I'll give you a moment to write it down if necessary, but basically it's the gist of the whole

#### lesson

, so here we go. to see what effect increasing certain

## dimensions

of a figure will have on elements of

, elements of

## dimensions

, so if we increase the radius by a factor of four, if the radius becomes four times larger, what What will happen to the diameter of a sphere?
Well if the diameter is one dimensional so it will become four times larger which will happen to the surface area of ​​the sphere well that is a 2d measurement so it will become four squared or 16 times larger, what will happen to the volume. be it four cubed or sixty four times as large the circumference what happens is the comfort of the sphere, well that's a measure of 1d so it will only be four times as large so the length, the width and height of a prism triple what h corresponds to the perimeter of the base perimeter which is a measure of 1d, so it will only triple.
What's going on? The area of ​​the surface that is a measure of 2d, so it will become three squares or 9 times bigger and the volume is a measure of 3d, so it will become three. cubed or twenty seven pounds bigger ok here i'm trying to give you a better idea of ​​what's really going on here so i have a sphere this is my original sphere it has a radius of our what happens if i double the radius well the radius and i I'm looking at the effect in volume, so the radius is doubled, but if you think of a formula where that radius is going to be converted to cubes, then the 2r is going to multiply by itself three times, basically, you'll see that in actually adds up to eight times bigger and what if I triple the radius well again you're tripling it in three directions three directions that affect the volume so three are cubes and that's why it's 27 times the volume okay so this is a classic problem that is going to be on the test and there are also quite a few of these in the homework they give you a one dimensional scale factor they say the figure is going to change because of this and then they ask what effect that will have on the surface area which is 2d or volume which is 3d so here they are saying the rectangular prism is shrinking by a scale factor of 1/4 so it is getting smaller now what is the 3d effect of that same scale factor well you will have cubed of the scale factor so it will be 1/4 cubed so basically the volume will be 1/64 of what it was so my original was the big one it was 200 cubic centimeters so for find the volume of the smaller I am going to compare the small with the big and I am going to establish a simple proportion.
I know my 3D scale factor is 1 to 64 and that would equal x over 200, that 200 has to go on the bottom because that's the volume of a larger figure I just cross multiply and solve and we get our answer 3.125 centimeters cubic now you should pause the video here and try this one so it gets bigger and bigger the scale factors 3 doesn't actually say it's a 1d scale factor in the problem, but if it just says the scale factor is 3, so you can assume it's 1d unless it says otherwise the 2d scale factor we're looking at in surface area here would be 9 this would be 3 squared so again the small was 40 that's the small .
I'm comparing big small to big again and my scale factor is 9 so the new 9 times bigger my ratio and in 2d it's going to be 1 to 9 like 40 is the ax forty has to go on top is it was the area of surface area of ​​the little one and I cross multiply and solve and we see that the new one, the bigger one, will have a surface area of ​​360 degrees, okay, the trickiest problems of all, in my opinion, are these: you get the ratio in 2d they give you surface area ratios this is a 2d measurement and then they ask for the ratio of their volumes now you could go from 2d to 3d in your head but i can't so i'm going na skip back to 1d if my 2d ratio is one to nine then my dear a she has to be the square root of that which would be 403 and now i can figure out what my 3d ratio is just by cube one two three and i get a 227 now if you can go from 2d to 3d in your head good for you let's do number two p To decalcify the factor 9 to 25 that would mean the scale factor 1d is the square root of that would be three to five and a 3d the scale factor would be the cube of 3 to 5 27 to 125 number three to DS four to nine Wendy would have to be two to three and then three D would be tequila which looks like it's 8 to 27 number four pretty much the same I think you get the idea here that's big and finally your bug of the day I have a good one our test is coming up and i hope you're ready for that i think i think you'll be fine
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