On a square cardboard sheet of area \(784\,\text{cm}^2\), four congruent circular plates of maximum size are placed such that each plate touches two others and each side of the square is tangent to two plates. Find the area of the square sheet not covered by the plates.
\(784 - 196\pi\,\text{cm}^2 \;\approx\; 168.25\,\text{cm}^2\)
Step 1: The square cardboard has an area of \(784\,\text{cm}^2\).
To find the side length of the square, take the square root of the area:
\(\text{Side of square} = \sqrt{784} = 28\,\text{cm}\).
Step 2: The four circular plates are placed inside the square so that each one touches two sides of the square and also touches two other circles. This means the circles are as large as possible within the square.
Step 3: Look at half of the side of the square: \(28/2 = 14\,\text{cm}\). Each circle will fit exactly in one corner, so the diameter of each circle = \(14\,\text{cm}\).
Step 4: Radius of each circle = half of diameter = \(14/2 = 7\,\text{cm}\).
Step 5: Area of one circle = \(\pi r^2 = \pi (7)^2 = 49\pi\,\text{cm}^2\).
Step 6: There are 4 circles. Total area of 4 circles = \(4 \times 49\pi = 196\pi\,\text{cm}^2\).
Step 7: Area of the square sheet not covered by the circles = Area of square – Total area of circles.
\(= 784 - 196\pi\,\text{cm}^2\)
Step 8: Approximate using \(\pi \approx 3.14\):
\(196 \times 3.14 \approx 615.75\)
Uncovered area = \(784 - 615.75 = 168.25\,\text{cm}^2\).
Final Answer: The uncovered area is \(168.25\,\text{cm}^2\).