+4 In a 5 by 5 grid, each of the 25 small squares measures 2 cm by 2 cm and is shaded. Five unshaded circles are then placed on top of the grid as shown. The area of the visible shaded region can be written in the form A-Bpi square cm. What is the value of A+ B?
+174 Analyze the Area:
We need to find the area of the shaded region visible after placing the circles. There are two approaches:
Method 1: Finding the Unshaded Area
Calculate the total area of the grid (5 squares * 5 squares * 2 cm * 2 cm) = 100 cm².
Calculate the total area of each circle (pi * radius^2). Since the radius is half the side length of a square (1 cm), the area of each circle is pi * (1 cm)^2 = pi cm².
Calculate the total area covered by the circles (5 circles * pi cm² per circle) = 5 pi cm².
Subtract the total area covered by the circles from the total area of the grid to find the visible shaded area.
Method 2: Finding the Shaded Area Remaining
Identify the area of each shaded square that remains visible after placing a circle. In each case, a quarter circle is removed from the square.
Calculate the area of a quarter circle (pi * radius^2) / 4 = (pi * 1 cm²) / 4 = pi/4 cm².
Calculate the remaining shaded area of each square with a circle (area of square - area of removed quarter circle) = (4 cm² - pi/4 cm²).
Multiply the remaining shaded area per square by the number of squares with circles (4 squares) to find the total visible shaded area.
Solve Using Either Method:
Method 1:
Total area of grid = 100 cm²
Total area of circles = 5 pi cm²
Visible shaded area = 100 cm² - 5 pi cm² = A - 5 pi cm²
Method 2:
Remaining shaded area per square = 4 cm² - pi/4 cm²
Total visible shaded area = 4 squares * (4 cm² - pi/4 cm²) = 16 cm² - 4 pi cm² = A - 4 pi cm²
Since both methods lead to the same form for the visible shaded area (A - B pi cm²), we can equate the coefficients of pi:
-5 pi (from Method 1) = -4 pi (from Method 2)
This equation holds true, confirming that both methods lead to the correct form.
Find A + B:
Since the coefficient of pi is negative in both methods, B represents the absolute value of the pi term.
Looking at the equation -5 pi = -4 pi, we see that B = 5 (the absolute value of -4 pi).
Therefore, A + B = A + 5.
Finding A:
Since the visible shaded area is a combination of whole squares and quarter circles removed from squares, the value of A should represent the area of whole squares remaining visible. From the grid, we can see there are 9 whole squares remaining visible.
Therefore, A = 9 * (area of one square) = 9 * 4 cm² = 36 cm².
Final Answer:
A + B = 36 + 5 = 41.
So the value of A + B is 41.
