Apoint in the figure selected at random. find the probability that the point will be in the part that is not shaded.

The probability that the point will be in the part that is NOT shaded is about 20% ⇒ 4th answer

Step-by-step explanation:

* Lets look to the figure

- There are four circles inscribed in  a square

- The four circles touched each other and touched the four

  sides of the square

∴ The side of the square = twice the diameter of a circle

- If the side of the square is l and the diameter of the circle is d

∴ l = 2d ⇒ divide the two sides by 2

∴ d = (1/2) l

∵ The radius of the circle = (1/2) the diameter

∴ r = (1/2) d

∵ d = (1/2) l

∴ r = (1/2)(1/2) l = (1/4) l

* Now lets find the area that NOT shaded

∵ The area of the square = side × side

∴ The area of the square = l × l = l²

∵ The area of the circle = πr²

∵ r = (1/4) l

∴ r² = [(1/4) l]² = (1/4)² × l² = (1/16) l²

∴ The area of one circle = (1/16)πl²

- The shaded part is the four circles

∴ The shaded area = 4 × (1/16)πl² = (1/4)πl²

- The part is not shaded = Area of the square - Area of the shaded part

∴ The area of not shaded = l² - (1/4)πl² ⇒ take l² as a common factor

∴ The area of not shaded = l²(1 -1/4 π)

- The probability that the point will be in the part that not shaded is

  area of the part not shaded/area of the square

∴ P = l²(1 - 1/4 π)/l² ⇒ cancel l² from up and down

∴ P = (1 - 1/4 π )/1 = 0.2146 ≅ 0.2

- Chang it to percent number

∴ P = 0.2 × 100% = 20%

* The probability that the point will be in the part that is NOT shaded

  is about 20%

∴ r = (1/2) d

∴

step-by-step explanation: