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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?

Apr 17, 2024

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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.