a 12x12-inch square is divided into n^2 congruent squares by equally spaced lines parallel to its sides. circles are inscribed in each of the squares. find the sum of the areas of the circles. answer and please explain step by step how u got the answer!! please!!!!
the diameter of each square is n, so the radius is n/2. The area of each circle is PI(n/2)^2
the area of all of them is that times n^2
total area circles n^4*PI/4
To find the sum of the areas of the circles inscribed in each of the congruent squares, we need to determine the number of congruent squares, and then calculate the area of each circle.
Step 1: Determine the number of congruent squares:
A 12x12-inch square is divided into n^2 congruent squares by equally spaced parallel lines. Since the width and height of the larger square are both 12 inches, the length of each division line will be 12/n inches.
To find the number of congruent squares, we can count the number of divisions along one side and then square it:
Number of squares = (12 inches) / (12/n inches) * (12 inches) / (12/n inches)
= n * n
= n^2
Step 2: Calculate the area of each circle:
The area of a circle can be calculated using the formula A = π*r^2, where A is the area and r is the radius.
In each congruent square, the circle will be inscribed, which means the diameter of the circle will be equal to the side length of the congruent square.
Since the congruent squares have side length 12/n inches, the diameter of each circle will be 12/n inches.
Therefore, the radius (r) will be half the diameter, so r = (12/n) / 2 = 6/n inches.
The area of each circle, A, can be calculated as A = π*(6/n)^2.
Step 3: Find the sum of the areas of the circles:
To find the sum of the areas of all the circles, we need to calculate the individual areas and sum them up.
Since there are n^2 congruent squares, the sum of the areas of the circles can be calculated as:
Sum of areas = n^2 * A
= n^2 * π*(6/n)^2
= n^2 * π*(6^2/n^2)
= π * 36
Therefore, the sum of the areas of the circles inscribed in each of the squares is 36π square inches.