answer:Given the triangle triangle ABC with side lengths opposite angles A, B, and C being a, b, and c respectively, and the ratio of these sides being a:b:c=3:sqrt{7}:2, we proceed as follows: 1. **Assign values to the sides based on the given ratio**: Let's express the sides in terms of a common variable t (t>0), such that a=3t, b=sqrt{7}t, and c=2t. This step ensures that the ratio of the sides is maintained while allowing us to apply the cosine rule. 2. **Apply the cosine rule to find cos B**: The cosine rule states that cos B = frac{a^2 + c^2 - b^2}{2ac}. Substituting the expressions for a, b, and c into this formula gives us: [ cos B = frac{(3t)^2 + (2t)^2 - (sqrt{7}t)^2}{2 cdot 3t cdot 2t} = frac{9t^2 + 4t^2 - 7t^2}{12t^2} = frac{6t^2}{12t^2} = frac{1}{2}. ] 3. **Determine the angle B**: Knowing that cos B = frac{1}{2} and considering that B is an angle in a triangle, thus B in (0, pi), we can determine that B = frac{pi}{3} because cos frac{pi}{3} = frac{1}{2}. 4. **Conclude the answer**: Since B = frac{pi}{3}, the correct answer to the measure of angle B is boxed{C: frac{pi}{3}}.

answer:Given the temperature conversion between Fahrenheit and Celsius scales, we know that: - The freezing point of water is 32^{circ} mathrm{F} = 0^{circ} mathrm{C}. - The boiling point of water is 212^{circ} mathrm{F} = 100^{circ} mathrm{C}. Step 1: Determine Conversion Factor From these, we get: 180^{circ} mathrm{F} = 100^{circ} mathrm{C} Thus, 1 degree Fahrenheit equates to: 1^{circ} mathrm{F} = frac{100^{circ} mathrm{C}}{180} = frac{5}{9}{}^{circ} mathrm{C} Step 2: Consider Maximum Deviation Due to Rounding in Fahrenheit When rounding the temperature to the nearest whole number in Fahrenheit, the maximum deviation can be: frac{1}{2}{}^{circ} mathrm{F} Which in Celsius is: frac{1}{2}{}^{circ} mathrm{F} times frac{5}{9}{}^{circ} mathrm{C} = frac{5}{18}{}^{circ} mathrm{C} Step 3: Consider Maximum Deviation Due to Rounding in Celsius When converting the rounded Fahrenheit temperature to Celsius and again rounding to the nearest Celsius whole number, the maximum deviation here is: frac{1}{2}{}^{circ} mathrm{C} However, due to the way Fahrenheit values correspond to Celsius values, the deviation in Celsius upon second rounding will be at most: frac{4}{9}{}^{circ} mathrm{C} Step 4: Add Up the Maximum Deviations Adding these maximum deviations, we get: frac{5}{18}{}^{circ} mathrm{C} + frac{4}{9}{}^{circ} mathrm{C} Let's combine the fractions: frac{5}{18} + frac{4}{9} = frac{5}{18} + frac{8}{18} = frac{13}{18}{}^{circ} mathrm{C} Conclusion Thus, the maximum possible deviation from the original temperature in Celsius, after rounding the temperature first in Fahrenheit and then in Celsius, is: boxed{frac{13}{18}{}^{circ} mathrm{C}}

answer:Firstly, simplify the fraction frac{66}{1110}. Since both numerator and denominator are divisible by 66, we get: frac{66}{1110} = frac{1}{16.81818181...} = frac{1}{frac{1110}{66}} = frac{66}{1110} = frac{1}{16.8overline{18}} We convert frac{1}{16.8overline{18}} back to fraction form to simplify the calculation, knowing that 16.8overline{18} as a fraction is frac{151}{9}. Thus, we have: frac{1}{16.8overline{18}} = frac{9}{151} Performing long division to find the decimal representation of frac{9}{151}: frac{9}{151} = 0.059602649... This decimal does not simplify readily to a clean repeating block, but upon further division, it appears to repeat with the sequence overline{059}. To find the 222nd digit after the decimal point, we consider the 221st digit after the first decimal point (since we're indexing from zero at the first digit). Since the repeat length is 3 digits, we compute: 221 mod 3 = 2 Thus, the 222nd digit is the second digit in the repeating block "overline{059}", which is boxed{5}.

answer:Given that (3^x cdot 4^y = 19,683) and (x = 9), we can substitute (x) into the equation to find (y). (3^9 cdot 4^y = 19,683) We know that (3^9 = 19,683), so we can simplify the equation to: (19,683 cdot 4^y = 19,683) To solve for (y), we divide both sides by (19,683): (4^y = frac{19,683}{19,683}) (4^y = 1) Since any number raised to the power of 0 is 1, we have: (y = 0) Now, we can find the difference between (x) and (y): (x - y = 9 - 0 = 9) So, the number that (x - y) equals is boxed{9} .