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... please answer and explain how you got this answer!!!
if there are n squares per side, each square is 12/n inches square.
For a square of side s, the inscribed circle has radius s/2.
So, the circles add up to
n^2 (pi (12/2n)^2) = pi n^2 36/n^2 = 36pi
Note that it is the same area as for a single inscribed 6-inch circle!
To find the sum of the areas of the circles inscribed in the squares, we need to determine the number of squares and the radius of each circle. Let's break down the problem step by step.
Step 1: Determine the number of squares.
One side of the 12x12-inch square is divided into n equal parts by the equally spaced lines. This means there are (n+1) lines parallel to each side. Therefore, there will be (n+1)^2 small congruent squares formed inside the larger square.
Step 2: Calculate the size of each small congruent square.
Since the length of one side of the large square is 12 inches, and it is divided into (n+1) segments, the length of each small congruent square will be 12/(n+1) inches. Since the small squares are congruent, their length and width are the same.
Step 3: Calculate the radius of each circle.
The circles are inscribed in each small congruent square, so the radius of each circle is half the length of the side of the square. Therefore, the radius of each circle will be (12/(n+1))/2 = 6/(n+1) inches.
Step 4: Calculate the area of each circle.
The formula for the area of a circle is A = πr^2. Substituting the radius we found in step 3, the area of each circle will be A = π(6/(n+1))^2.
Step 5: Calculate the sum of all circle areas.
Since there are (n+1)^2 small congruent squares, there will be (n+1)^2 circles as well. To find the sum of their areas, we can multiply the area of each circle found in step 4 by the number of circles and sum them up:
Sum of Circle Areas = (n+1)^2 * π(6/(n+1))^2
= (n+1) * π * (6/(n+1))^2
= π * 36/(n+1)
Therefore, the sum of the areas of the circles is equal to π * 36/(n+1).