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Bunuel
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kartikeysharma
I'm sorry, but the solution is still not clear to me. How is the answer A and not C?


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? 🙂

Posted from my mobile device
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Bunuel
After a one-piece circular tablecloth of maximum possible area was cut from a rectangular piece of material, what was the number of square inches of material that remained?

(1) The piece of material was 48 inches wide and 56 inches long before the tablecloth was cut.
(2) The radius of the tablecloth was 24 inches.


DS20411
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.
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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.
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