If a quarter circle is placed on a square target and the radius of the quarter circle is the same as the length of a side of the square, what is the probability that a dart that hits the square does not hit the quarter circle
Prob= success area/total area= (areasquare-areaquartercirlce)/area square.
To find the probability that a dart hitting the square does not hit the quarter circle, we need to compare the areas of the square and the quarter circle.
First, let's calculate the areas:
Area of the square = (side length)^2
Area of the quarter circle = (1/4) * π * (radius)^2
In this case, the side length of the square is the same as the radius of the quarter circle. Let's represent this length as "r" for simplicity.
So, the area of the square = r^2
And the area of the quarter circle = (1/4) * π * r^2
Now, to find the probability that a dart hitting the square does not hit the quarter circle, we need to divide the area of the square that does not intersect with the quarter circle by the total area of the square.
The area of the square that does not intersect with the quarter circle is the difference between the area of the square and the area of the quarter circle:
Area of the square without the quarter circle = Area of the square - Area of the quarter circle
= r^2 - (1/4) * π * r^2
= r^2 - (π/4) * r^2
Therefore, the probability that a dart hitting the square does not hit the quarter circle can be calculated by dividing the area of the square without the quarter circle by the area of the square:
Probability = Area of the square without the quarter circle / Area of the square
= (r^2 - (π/4) * r^2) / r^2
= (1 - π/4)
Hence, the probability that a dart hitting the square does not hit the quarter circle is approximately 1 - (π/4), or about 0.2146.