First you need to find the area of the circle, which is πr^2
With a radius of 8 inches, the area of the circle will be π*8^2 or 64π.
Next you want to find the area of the 120 degree section of that circle.
120 degrees is 1/3 of a circle, so all you have to do next is multiply the are of the circle by 1/3 (because 360/120= 1/3).
(1/3)(64π) --> All you really have to do here is multiply 64 by 1/3 or divide it by 3 (because to divide by a number is the same a multiplying by it's reciprocal) so you get (64/3)(π).
64/3 is 21 and 1/3 or 21.333..., so however you choose to write it, I will leave the answer as (64/3)π squared inches.
Apples went up 0.75 a lb
p is the start price so new price is p + 0.75
we buy 3 lbs
3 ( p+ 0.75) = 5.88
now solve
3p + 2.25 = 5.88
-2.25 -2.25
3p = 3.63
/3 /3
p = 1.21 a lb
2^3 * 2^-5 = 2^(3 + (-5) = 2^(3 - 5) = 2^-2
A irrational number is a number that can't be expressed as a ratio of two whole numbers. That's it.
For examples (in increasing order of difficulty)
1 is a rational number because it is 1/1
0.75 is a rational number because it is equal to 3/4
2.333... (infinite number of digits, all equal to three) is rational because it is equal to 7/3.
sqrt(2) is not a rational number. This is not completely trivial to show but there are some relatively simple proofs of this fact. It's been known since the greek.
pi is irrational. This is much more complicated and is a result from 19th century.
As you see, there is absolutely no mention of the digits in the definition or in the proofs I presented.
Now the result that you probably hear about and wanted to remember (slightly incorrectly) is that a number is rational if and only if its decimal expansion is eventually periodic. What does it mean ?
Take, 5/700 and write it in decimal expansion. It is 0.0057142857142857.. As you can see the pattern "571428" is repeating in the the digits. That's what it means to have an eventually periodic decimal expansion. The length of the pattern can be anything, but as long as there is a repeating pattern, the number is rational and vice versa.
As a consequence, sqrt(2) does not have a periodic decimal expansion. So it has an infinite number of digits but moreover, the digits do not form any easy repeating pattern.
