Cindy is baking a 2-layer cake. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 10 inches. What
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72 
1÷9 = 111.... 2÷9 = 222.... and so on.
The proof is a little more difficult.
Think of all those repeating ones as a variable - let's call it n
so n= .111111......... repeating
How can we get a single one of those ones to jump across the decimal and be on the left side. We can multiply all of those ones by 10.
10n (ten times the original number) = 1.1111111 (ones still go on forever)
Now here is the interesting part. Let's take all the repeating ones in the first number we made away from the second number.
10n = 1. 1111111......
<u>- n = . 1111111....
</u>9n = 1 (all of the repeaters are gone and only the one we moved to the left
of the decimal is left)
Now let's divide by 9 to get n by itself
<u>9n</u> = <u>1
</u>9 9
And voila! n = 1/9
So to repeat 72.111... written as a fraction is 72
The proof is a little more difficult.
Think of all those repeating ones as a variable - let's call it n
so n= .111111......... repeating
How can we get a single one of those ones to jump across the decimal and be on the left side. We can multiply all of those ones by 10.
10n (ten times the original number) = 1.1111111 (ones still go on forever)
Now here is the interesting part. Let's take all the repeating ones in the first number we made away from the second number.
10n = 1. 1111111......
<u>- n = . 1111111....
</u>9n = 1 (all of the repeaters are gone and only the one we moved to the left
of the decimal is left)
Now let's divide by 9 to get n by itself
<u>9n</u> = <u>1
</u>9 9
And voila! n = 1/9
So to repeat 72.111... written as a fraction is 72
Step-by-step explanation:
Take the natural log of both sides:
Logarithm rules allow you to bring down the exponents:
Now differentiate. We will have to implicitly differentiate 'y' since it is a function of 'x'. Both sides require the product rule:
Isolate the terms that have y' since that is what we want:
Factor out y' to get:
Therefore: