Answer:
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
∴
