If a dart is thrown at a square target with a circle inscribed in it such that each side of the square is tangent to the circle, what is the approximate probability that the ball will hit inside the circle?
That depends upon the skill of the dart thrower. One would not expect a random distribution over the entire square area of the target.
If the distribution were random and uniform over the entire area, the answer would be pi/4.
To find the approximate probability that the dart will hit inside the circle, we need to compare the areas of the circle and the square.
First, we need to determine a relationship between the circle and the square. Since each side of the square is tangent to the circle, we know that the diagonal of the square is equal to the diameter of the circle.
Let's assume the side length of the square is 's'. Since the diagonal of a square divides it into two congruent right triangles, we can use the Pythagorean theorem to find the relationship between the side length of the square and the diameter of the circle:
Diagonal^2 = Side^2 + Side^2
diameter^2 = s^2 + s^2
diameter^2 = 2s^2
diameter = sqrt(2s^2)
diameter = s * sqrt(2)
Now, let's consider the areas of the circle and the square.
The area of the circle is given by:
Area_circle = pi * (radius)^2
Since the diameter of the circle is equal to s * sqrt(2), the radius is half of the diameter:
radius = (s * sqrt(2))/2
Therefore, the area of the circle is:
Area_circle = pi * [ (s * sqrt(2))/2 ]^2
The area of the square is given by:
Area_square = side^2
Area_square = s^2
Now, we can calculate the approximate probability:
Probability = Area_circle / Area_square
Probability = [ pi * (s * sqrt(2))/2^2 ] / s^2
Probability = ( pi * s^2 * 2/4 ) / s^2
Probability = pi/4
So, the approximate probability that the dart will hit inside the circle is pi/4, which is approximately 0.785.
Please note that this is an approximation and assumes a uniform distribution of dart throws within the square.