doubt in geometry.?

Question

a rectangular sheet has its length equal to twice its breadth. it is cut into two equal square sheets. the first square sheet is cut into exactly n small squares, not necessarily of the same size. from each small square, the maximum possible circle is cut. frim the second square sheets, the max. possible circle is cut. the parts of the sheets which remain are discarded. find the ratio of the sum of the areas of all the circles cut from the first square sheet and the area of the circle cut from the second square sheet.

options :
1:1
1:2
2:1
4:1
depend on the value of n

please explain this answer to me.

Answer 1

Cutting the "maximum possible" circle from a square will always give an area of pi/4 (approximately 0.785) times the original square's area.

(The area of a square whose side is x is x^2 (x squared).
The area of the "maximum possible" circle, inscribed within that square, is pi*r^2 (here the radius r is x/2), which equals pi*(x/2)^2 = (pi/4)*(x^2).)

Call the area of each square sheet A. Call the areas of the n small squares a[1], a[2], a[3], ..., a[n]. The sum of the small squares' areas is A. "Cut into exactly n small squares" implies that there's no "leftover" area.

The total area of all the inscribed circles from the small squares is (pi/4)*a[1] + (pi/4)*a[2] + ... + (pi/4)*a[n].
This equals (pi/4)*( a[1]+a[2]+...+a[n] ), which equals (pi/4)*A, which is also the area of the single inscribed circle from the second square sheet.

So the ratio is 1:1.

Answer 2

The area of a circle inscribed in a square is π/4 * area of the square. (If a square is d by d, the area is d^2; the diameter of the inscribed circle is d, so its area is π/4* d^2. The ratio is then π/4.)

The area of the n circles is ∑(1 to n)*π/4*An, where An is the area of square n. Since π/4 is a constant, this expression is equivalent to π/4 * ∑(1 to n)*An. This summation is merely the area of the original square paper, A so the result is π/4 * A.

On the other side, the area in the circle is π/4 * area of the square, which is A, or π/4 * A, the same as above. So the answer is 1:1

Answer 3

it will be 1:1 or 1:2 or 2:1 or 4:1

Answer 4

1:1

Answer 5

one easy thing
u said l = 2b
when it is cut into small squares
then both will be equal
automatically area of circles will be equal
it cant be 1:1 or 2:1 or 4:1 as they r same length sqaures
so it is 1:1 just logically

Answer 6

inscribed circle /square is a constant for all pairs therefore,
amount circle left is same in each case. So 1 : 1

Answer 7

Area of a square of side (a) = a2 -------- (1)

Area of a circle with diameter (a) = (Pi/4)*a2 --------- (2)

No matter, how many number of pieces you may cut a square into, the area remains the same.
i.e., If you cut a square (of side a) into n number of pieces (of sides a1,a2,.....an), then
a12+a22+a32+a42+.........+an2 = a2 -------- (3)

The Diameter of the maximum possible circle that can be cut from a square of side (a) is (a). ----------- (4)

Coming to this particular problem, when you cut a rectangle (which has its length equal to twice its breadth) into two equal squares, the side of each square will be equal to breadth of the rectangle, let us assume it to be (a) -------- (5)

Now assume that the first square is cut into (n) no of smaller squares of sides a1,a2,a3,..........an.

From eq(4), we can derive that the diameters of the maximum possible squares cut from each square are a1,a2,a3..............an.

Sum of areas of all these squares
= (pi/4)*a12+(pi/4)*a22+(pi/4)*a32+........+)pi/4)*an2
= (pi/4)*(a12+a22+a32+......an2)
= (pi/4)*a2 ------- from eq(3)

So, sum of areas of all smaller squares cut from Square 1 = (pi/4)*a2 --------- (6)

Now from eq(4), we can derive that the diameter of the maximum possible circle cut from the square 2 = a (side of the square).

So, area of the circle cut from square 2 = (pi/4)*a2 ----- (7)

So required ratio = (6) : (7) = (pi/4)*a2 : (pi/4)*a2 = 1 : 1

So, Irrespective of the no of smaller squares that the square 1 is cut into, the ration remains the same, which is 1 : 1.