A square napkin is folded in half on the diagonal and placed on the diameter of a round plate (see diagram below). If the folded napkin spans the plate and the plate has an 18-inch diameter, what is the area of the space on the plate that is NOT covered by the napkin? Leave your answer in terms of .

Step-by-step explanation:

step 1

Find the area of the plate

The area of a circle is given by the formula

we have

---> the radius is half the diameter

substitute

step 2

Find the area of the square napkin folded (is a half of the area of the square napkin)

we know that

The diagonal of the square is the same that the diameter of the plate

Applying Pythagorean theorem

where

b is the length side of the square

we have

substitute

solve for b^2

-----> is the area of the square

Divide by 2

step 3

Find the area of the space on the plate that is NOT covered by the napkin

we know that

The  area of the space on the plate that is NOT covered by the napkin, is equal to subtract the area of the square napkin folded (is a half of the area of the square napkin) from the area of the plate

so

simplify

1,645,493,059.35

step-by-step explanation:

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