answer:1. **Let the total number of students be n:** - Number of juniors = 0.2n - Number of seniors = 0.8n 2. **Calculate the total score using class average:** - Total class score = Class average x Total number of students = 84n 3. **Calculate total score of seniors:** - Total senior score = Senior average x Number of seniors = 82 times 0.8n = 65.6n 4. **Determine total junior’s score from total scores:** - Total junior score = Total class score - Total senior score = 84n - 65.6n = 18.4n 5. **Compute each junior's score:** - Score per junior = Total junior score / Number of juniors = frac{18.4n}{0.2n} = 92 6. **Conclusion**: Each junior received a score of 92 on the test. The final answer is boxed{textbf{(C) } 92}

answer:To find the area of a trapezium (also known as a trapezoid), you can use the following formula: Area = (1/2) * (Sum of the lengths of the parallel sides) * (Distance between the parallel sides) In this case, the lengths of the parallel sides are 26 cm and 18 cm, and the distance between them is 15 cm. Let's plug these values into the formula: Area = (1/2) * (26 cm + 18 cm) * 15 cm Area = (1/2) * 44 cm * 15 cm Area = 22 cm * 15 cm Area = 330 cm² So, the area of the trapezium is boxed{330} square centimeters.

answer:1. First, we need to simplify the expressions inside the logarithms and analyze the properties of the quadratic function f(x) which opens downward and satisfies f(2-x) = f(2+x) for all x. 2. Consider the expression x^2 + x + frac{1}{2}: [ x^2 + x + frac{1}{2} = left(x + frac{1}{2}right)^2 + frac{1}{4} geq frac{1}{4} ] Since x^2 + x + frac{1}{2} is inside the logarithm base frac{1}{2}, we have: [ log_{frac{1}{2}}left(x^2 + x + frac{1}{2}right) leq log_{frac{1}{2}}left(frac{1}{4}right) = 2 ] 3. Similarly, consider the expression 2x^2 - x + frac{5}{8}: [ 2x^2 - x + frac{5}{8} geq frac{5}{8} text{ for all } x ] Here, we seek to find log_{frac{1}{2}}left(2x^2 - x + frac{5}{8}right): [ log_{frac{1}{2}}left(2x^2 - x + frac{5}{8}right) leq log_{frac{1}{2}}left(frac{5}{8}right) = 1 ] 4. Given that f(2-x) = f(2+x), the function has its axis of symmetry at x = 2. Since f(x) opens downward, f(x) is decreasing when x > 2 and increasing when x < 2. 5. We need to solve the inequality: [ fleft[log_{frac{1}{2}}left(x^2 + x + frac{1}{2}right)right] < fleft[log_{frac{1}{2}}left(2x^2 - x + frac{5}{8}right)right] ] Because f(x) is decreasing for x > 2 and logarithms also reverse the inequality due to the change of base property: [ log_{frac{1}{2}}left(x^2 + x + frac{1}{2}right) < log_{frac{1}{2}}left(2x^2 - x + frac{5}{8}right) ] 6. Solve the inequality inside the logarithms: [ x^2 + x + frac{1}{2} < 2x^2 - x + frac{5}{8} ] Simplify it: [ x^2 + x + frac{1}{2} - 2x^2 + x - frac{5}{8} < 0 ] [ -x^2 + 2x - frac{1}{8} < 0 ] 7. Solve the quadratic inequality: [ x^2 - 2x + frac{1}{8} > 0 ] Use the quadratic formula for roots: [ x = frac{2 pm sqrt{4 - frac{1}{2}}}{2} = frac{2 pm sqrt{frac{7}{2}}}{2} = frac{2 pm frac{sqrt{14}}{2}}{2} ] [ x = 1 pm frac{sqrt{14}}{4} ] 8. The solution to the inequality: [ 1 - frac{sqrt{14}}{4} < x < 1 + frac{sqrt{14}}{4} ] Conclusion: [ boxed{1 - frac{sqrt{14}}{4} < x < 1 + frac{sqrt{14}}{4}} ]

answer:1. **Calculate sum of ages in the first room:** The average age of the 8 people in the first room is 20 years old. Therefore, the total sum of their ages is: [ 8 cdot 20 = 160 text{ years} ] 2. **Calculate sum of ages in the second room:** The average age of the 12 people in the second room is 45 years old. Thus, the total sum of their ages is: [ 12 cdot 45 = 540 text{ years} ] 3. **Find the total average age before anyone leaves:** The combined sum of ages in both rooms is: [ 160 + 540 = 700 text{ years} ] And the total number of people is: [ 8 + 12 = 20 ] Therefore, the average age of all the people before anyone leaves is: [ frac{700}{20} = 35 text{ years} ] 4. **Formulate the new average after one person leaves:** Let ( x ) be the age of the person who leaves. With the departure of this person, the total number of people becomes ( 20 - 1 = 19 ). The new sum of ages is: [ 700 - x ] The average age after the departure increases by 1 year, making it: [ 35 + 1 = 36 text{ years} ] 5. **Set up the equation and solve for ( x ):** Based on the new conditions, we set up the equation: [ frac{700 - x}{19} = 36 ] Solving for ( x ): [ 700 - x = 36 cdot 19 ] Calculate ( 36 cdot 19 ): [ 36 cdot 19 = 684 ] Thus: [ 700 - x = 684 ] Therefore: [ x = 700 - 684 = 16 ] Conclusion: The age of the participant who left is ( boxed{16} ).