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

Question

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

nick1816 · Answer

To maximize the area of the circle, diameter of the cloth = d = min(a,b) , where a and b are the sides of the rectangular piece

Area of remaining portion =  a*b - \frac{\pi*d^2}{4 }

Statement 1=

We know a and b; hence we can find d.

Sufficient

Statement 2-

We know d; hence we can find the shorter side of the rectangle. We have no idea about the longer side.

Insufficient

kartikeysharma · Answer

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

Fdambro294 · Answer

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

Tableclothlinen · Answer

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 	imes 56 = 2688 	ext{ 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 	imes 24^2 = 576\pi 	ext{ 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 	imes 3.1416 \approx 1809.6 2688−1809.6≈878.4 square inches2688 - 1809.6 \approx 878.4 	ext{ square inches}   
So, about  878 square inches of material remained  after cutting the largest possible circular tablecloth.

Tableclothlinen · Answer

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: 
  
So, about 878 sq in of material remained.

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

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