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Current time:0:00Total duration:6:32

Video transcript

in the last video we got as far as figuring out the area of this Coach snowflake this thing that has an infinite perimeter can be expressed as this as this infinite sum right over here and so our job in this video is to try to simplify this and hopefully get a finite value so let's do our best to actually simplify this thing right over here so the easiest part of this thing to simplify is this right over here so let's just focus on that and then if we can get a value for this part that I'm bracketing off then we can just place that value here and then simplify the rest of it so what I just bracket avout what I've just bracketed off can be re-written as 3 times 4/9 3 times 4/9 plus 4/9 squared 4 9 squared plus 4/9 to the third power to the third power third power and you can go on and on and on well plus 4 nights to every other power all the way through infinity and lucky for us there is a way to figure out this infinite what is called a geometric series there's a way to figure this out and there's a we do videos actually I think I've done several videos where we proved the general thing but I'll just do it by hand this time just so that we don't have to resort to some magical formulas so let's say that we define some sum let's say that we say this sum right over here let's call that s so I'm going to say that s s is equal to what we have in parentheses right over here it's going to be equal to 4/9 plus 4/9 squared plus 4/9 to the third power for a nice to the 3rd power all the way to infinity now let's also say that let's multiply s times 4/9 so let me write that down here so if we multiply s times 4/9 let me give myself some space here so what's 4/9 s going to look like so 4 9s so then I'm just essentially multiplying every term here times 4/9 so if I take this first term and multiply it times 4/9 what am I going to get well I'm going to get 4/9 squared 4/9 squared if I take the second term and multiply it times 4/9 I'm going to get I'm going to get 4/9 to the third power and we're going to go all the way to infinity so this is interesting when I multiply four nice times s I get all of the terms here except for this first four ninths and now this is kind of the magic of how we can actually find the sum of an infinite an infinite geometric series is we can subtract this term right over here we can subtract this pink line from this green line so if we do that if we do that clearly this is equal to that and that is equal to that so if we subtract this from that it's equivalent to subtracting the pink from the green so we get s minus 4/9 sorry s minus 4/9 s minus four let me do that in that pink color just so we know what we're doing minus 4/9 s is equal to well every other term this guy - this guy is going to cancel out this guy - this guy is going to cancel out and that's going to happen all the way all the way to infinity and on the right hand side you're only going to be left with you're only going to be left with this 4/9 right over here only going to be left with this four Knights and then this 4/9 we can s is the same thing as nine over nine so we could write this as 9 over 9 s minus 4/9 s s is equal to 4/9 and so 9 over 9 minus 4 over 9 of something gives us 5 over 9 so this becomes five ninths s is equal to 4/9 and then to solve for s and this is kind of magical but it's actually quite logical multiply both sides times the inverse of this so times nine fifths so that we can isolate the s times nine fifths these guys cancel out and we get s is equal to 4/5 s is equal to 4/5 that's really this is this is kind of neat this whole thing right over here we have just shown to ourselves is equal to 4/5 is equal to 4/5 so this entire bracket that we did right over here is equal to three times 4/5 so this entire bracket right over here is equal to three times 4/5 or it's equal to 12 to equal to 12 over five that entire bracket so now let's go to our original expression just so we don't lose track of what we're doing we have the square root of 3 square root of 3 s squared over 16 and then we have this 4 here 4 Plus this whole thing this whole thing right over here simplified to twelve fifths plus 12 plus twelve fifths now just to add these two things we can rewrite four is twenty fifths we can rewrite this as twenty we can rewrite this as 20 over five and then 20 over five plus 12 over 5 is 32 over five let me write this down right over here so this whole thing right over here is going to be 32 32 over five and this is really the homestretch now this is very exciting we're about to find the finite area of something that has an infinite perimeter so it's going to be let me rewrite it just I don't want to get messy in all the excitement here square root of 3 s squared over 16 times 32 over five times thirty-two over five and 32 we can divide both the numerator and the denominator by 16 32 divided by 16 is 2 16 divided by 16 is 1 and we are left with and this is where we really do need a drumroll the area of a of a Coach snowflake where the initial equilateral triangle that we start with has each of its sides is length s on there all length s because an equilateral triangle is I'll do this in magenta square root was going to be 2 times the square root of 3/2 times the square root of 3 s squared so whatever that side length was all of that over let's see we use the 2 we use the square root of 3 we use the s squared all of that over all of that over 5 so for example if that first equilateral triangle if that first equilateral triangle we started out with had a side length of 1 then the area of this crazy thing that has an infinite perimeter would just be 2 square roots of 3 times 1 squared over 5 or 2 square roots of 3 over 5 anyway I think that's that's kind of kind of cool