answer:1. **Understanding the Problem**: A circle with radius ( r ) and a concentric square of side ( 3 ) are given. We are to find ( r ) such that the probability of seeing exactly two sides of the square from a random point on the circle is ( frac{1}{2} ). 2. **Defining visible arcs**: From any point on the circle, sections will allow the view of exactly two sides of the square. Sides are seen entirely when viewed from arcs subtended by diagonals emanating from the circle's center to square's vertices. 3. **Probability and Arc Calculation**: The total measure of these arcs must be ( 180^circ ) because the probability condition translates to half of the circle's circumference. With a square, this means each relevant arc is ( frac{180^circ}{4} = 45^circ ). 4. **Triangle Analysis**: Let ( O ) be the circle's center and ( A, B ) be endpoints of an arc between adjacent vertices of the square. Triangle ( OAB ) forms an isosceles triangle with ( angle AOB = 45^circ ), hence ( angle OAB = 67.5^circ ) (since ( angle OAB ) is half of ( angle AOB )). 5. **Trigonometric Calculation**: Using the sine rule, ( sin 67.5^circ = frac{sqrt{6} + sqrt{2}}{4} = frac{AB/2}{r} ), where ( AB ) is the diagonal of the square, ( AB = 3sqrt{2} ). Therefore: [ frac{sqrt{6} + sqrt{2}}{4} = frac{3sqrt{2}/2}{r} implies r = frac{3sqrt{2}}{frac{sqrt{6} + sqrt{2}}{4}} = left(frac{4 times 3sqrt{2}}{sqrt{6} + sqrt{2}}right) ] Simplifying further gives ( r = 6sqrt{2} - sqrt{2} ). [ 6sqrt{2-sqrt{2}} ] boxed{The correct answer is B) ( 6sqrt{2}-sqrt{2} ).}

answer:To find out how many boys belong to other communities, we first need to calculate the total percentage of boys who are Muslims, Hindus, and Sikhs. Then we subtract this from 100% to find the percentage of boys from other communities. Percentage of Muslims: 44% Percentage of Hindus: 28% Percentage of Sikhs: 10% Total percentage of Muslims, Hindus, and Sikhs: 44% + 28% + 10% = 82% Now, we subtract this from 100% to find the percentage of boys from other communities: 100% - 82% = 18% Now, we calculate 18% of the total number of boys in the school to find out how many belong to other communities. Number of boys in the school: 850 Percentage of boys from other communities: 18% Number of boys from other communities = 18% of 850 = (18/100) * 850 = 0.18 * 850 = 153 Therefore, boxed{153} boys belong to other communities.

answer:To find the probability of drawing two red balls from the bag, we can use the formula for combinations to determine the number of ways to choose 2 red balls from 4, and divide it by the number of ways to choose any 2 balls from the total of 6 balls. The number of ways to choose 2 red balls from 4 is given by the combination formula: C_{4}^{2} = frac{4!}{2!(4-2)!} = frac{4 times 3}{2 times 1} = 6 Similarly, the number of ways to choose any 2 balls from the total of 6 is given by: C_{6}^{2} = frac{6!}{2!(6-2)!} = frac{6 times 5}{2 times 1} = 15 Therefore, the probability of drawing two red balls is: frac{C_{4}^{2}}{C_{6}^{2}} = frac{6}{15} = boxed{frac{2}{5}} So the correct answer is A. This problem involves the application of the combination formula to calculate probabilities, and is a basic question in probability theory.

answer:We'll use the Shoelace Theorem to calculate the area of the quadrilateral. The vertices in order are: (0,0), (2,4), (6,0), and (2,6). According to the theorem: 1. Calculate the sum of products of coordinates going down and to the right: [ 0 cdot 4 + 2 cdot 0 + 6 cdot 6 + 2 cdot 0 = 0 + 0 + 36 + 0 = 36 ] 2. Calculate the sum of products of coordinates going down and to the left: [ 0 cdot 2 + 4 cdot 6 + 0 cdot 2 + 6 cdot 0 = 0 + 24 + 0 + 0 = 24 ] 3. Subtract the second sum from the first sum: [ 36 - 24 = 12 ] 4. Multiply by frac{1}{2} to find the area: [ text{Area} = frac{1}{2} times 12 = 6 ] Therefore, the area of this quadrilateral is 6. The final answer is The correct answer given the choices is boxed{(C) 6}.