1.
A punching machine is used to punch a circular hole of diameter two Units from a square sheet of aluminium of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square:The proportion of the sheet area that remains after punching is:
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By: anil on 05 May 2019 02.29 am
The area of triangle ABC is 1/2 * $$sqrt2$$ * $$sqrt2$$ = 1
Area of semi-circle ABC = $$pi/2$$
So, area of circle outside the square = $$pi/2$$ - 1 = ($$pi$$ -2)/2
So, area of circle inside the sheet = $$pi$$ - ($$pi/2$$ - 1) = 1 + $$pi/2$$
Area of original square = 2*2 = 4
So, area of the sheet after punching = 4 - 1 - $$pi/2$$ = 3 - $$pi/2$$
So, proportion of sheet that remains after punching = (3 - $$pi/2$$)/4 = (6 - $$pi$$)/8
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The area of triangle ABC is 1/2 * $$sqrt2$$ * $$sqrt2$$ = 1
Area of semi-circle ABC = $$pi/2$$
So, area of circle outside the square = $$pi/2$$ - 1 = ($$pi$$ -2)/2
So, area of circle inside the sheet = $$pi$$ - ($$pi/2$$ - 1) = 1 + $$pi/2$$
Area of original square = 2*2 = 4
So, area of the sheet after punching = 4 - 1 - $$pi/2$$ = 3 - $$pi/2$$
So, proportion of sheet that remains after punching = (3 - $$pi/2$$)/4 = (6 - $$pi$$)/8