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Answer:
Area of the shaded part is 3.14 square unit.
Step-by-step explanation:
Area of the shaded part = Area of large semicircle - (Area of two small semi circles)
Area of large semicircle with center O =
=
= 2π
Area of semicircle with center O' =
=
Area of semicircle with center O" =
Now substitute these values in the formula,
Area of shaded part =
=
= 3.14 square unit
Area of the shaded part is 3.14 square unit.
Sum/difference:
Let
This means that
Now, assume that is rational. The sum/difference of two rational numbers is still rational (so 5-x is rational), and the division by 3 doesn't change this. So, you have that the square root of 8 equals a rational number, which is false. The mistake must have been supposing that
was rational, which proves that the sum/difference of the two given terms was irrational
Multiplication/division:
The logic is actually the same: if we multiply the two terms we get
if again we assume x to be rational, we have
But if x is rational, so is -x/15, and again we come to a contradiction: we have the square root of 8 on one side, which is irrational, and -x/15 on the other, which is rational. So, again, x must have been irrational. You can prove the same claim for the division in a totally similar fashion.