After a one-piece circular tablecloth of maximum possible area was cut

We only know that the figure is a rectangle.
We don’t know whether the rectangle is a square or not.

The circle we can draw within a rectangle will be limited by the shorter side (here the circle must stay within the boundaries of the rectangle)

Imagine a circle with diameter 4.

The MAX Area circle we could draw would be perfectly inscribed within a square with equal side lengths of 4.

Now imagine the square is stretched along its length.

We now have a rectangle with width = 4 and the length = 100

Again, the MAX circle we can draw within this rectangle will have a diameter of 4: the same circle as in the case of the square with side length 4.

The circle is a set of equidistant points from the center. Hence, no matter how we tried to draw the circle, we are limited by the shorter side of the rectangle.

Hope that helps a bit? 🙂

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Let's break it down step by step:

• From Statement (1), the original rectangular material was 48 inches wide and 56 inches long. That gives an initial area of:
  48×56=2688 square inches48 \times 56 = 2688 \text{ square inches}
• From Statement (2), the radius of the circular tablecloth is 24 inches, which means its diameter is 48 inches (so it fits within the width of the fabric).
  
  • The area of the circular tablecloth is: π×242=576π square inches\pi \times 24^2 = 576\pi \text{ square inches}
• To find the remaining material, subtract the area of the circular tablecloth from the original fabric:
  2688−576π2688 - 576\pi
  Since π\pi is approximately 3.1416, we get:
  576×3.1416≈1809.6576 \times 3.1416 \approx 1809.6 2688−1809.6≈878.4 square inches2688 - 1809.6 \approx 878.4 \text{ square inches}

So, about 878 square inches of material remained after cutting the largest possible circular tablecloth.

The original fabric was 48 × 56 = 2688 sq in. The circular tablecloth, with a 24-inch radius, had an area of 576π ≈ 1809.6 sq in.

Remaining material:
[2688 - 1809.6 \approx 878 \text{ sq in} ]
So, about 878 sq in of material remained.