answer:① There is exactly one line through a point outside a plane that is perpendicular to the plane, which is correct; ② There is exactly one line through a point outside a plane that is parallel to the plane, which is incorrect. There are infinitely many lines parallel to the plane; ③ If two parallel planes intersect a third plane, then the two lines of intersection are parallel, which is correct, according to the properties of parallel planes; ④ If two planes are perpendicular to each other, then a line passing through a point in the first plane and perpendicular to the second plane must lie within the first plane, which is correct, according to the properties of perpendicular planes. Therefore, the answer is: boxed{①③④}

answer:Let's denote the first term of the arithmetic sequence as a_1 and the common difference as d. We can use the formula for the sum of the first n terms of an arithmetic sequence, which is given by: S_n = frac{n}{2} [2a_1 + (n-1)d] For the first 5 terms, the sum S_5 is 10: S_5 = frac{5}{2} [2a_1 + (5-1)d] = 10 For the first 10 terms, the sum S_{10} is 50: S_{10} = frac{10}{2} [2a_1 + (10-1)d] = 50 Let's solve these two equations to find a_1 and d. For S_5: 10 = frac{5}{2} (2a_1 + 4d) Rightarrow 4 = 2a_1 + 4d Rightarrow 2a_1 + 4d = 4 ldots (1) For S_{10}: 50 = frac{10}{2} (2a_1 + 9d) Rightarrow 10 = 2a_1 + 9d Rightarrow 2a_1 + 9d = 10 ldots (2) Subtracting equation (1) from equation (2): (2a_1 + 9d) - (2a_1 + 4d) = 10 - 4 5d = 6 d = frac{6}{5} Substitute d into equation (1) to find a_1: 2a_1 + 4left(frac{6}{5}right) = 4 2a_1 + frac{24}{5} = frac{20}{5} 2a_1 = -frac{4}{5} a_1 = -frac{2}{5} Now to find the sum of the first 15 terms, S_{15}: S_{15} = frac{15}{2} [2a_1 + (15-1)d] S_{15} = frac{15}{2} [2left(-frac{2}{5}right) + 14left(frac{6}{5}right)] S_{15} = frac{15}{2} left[-frac{4}{5} + frac{84}{5}right] S_{15} = frac{15}{2} times frac{80}{5} S_{15} = 15 times 8 S_{15} = 120 Therefore, the sum of the first 15 terms is boxed{120}.

answer:Start by applying the Cauchy-Schwarz inequality in a similar fashion to the original problem: [ (a^2 + b^2 + c^2)(x^2 + y^2 + z^2) ge (ax + by + cz)^2. ] Substituting the given values: [ 49 cdot 64 ge 56^2. ] Calculate: [ 3136 ge 3136. ] We achieve equality, indicating that: [ frac{a}{x} = frac{b}{y} = frac{c}{z}. ] Let k = frac{a}{x} = frac{b}{y} = frac{c}{z}. Then, a = kx, b = ky, c = kz, and hence: [ a^2 + b^2 + c^2 = k^2(x^2 + y^2 + z^2). ] Substitute the sums of squares: [ 49 = k^2 cdot 64, ] which simplifies to k^2 = frac{49}{64}, and thus: [ k = frac{7}{8}. ] Therefore, the ratio (a+b+c)/(x+y+z), which equals k, is: [ boxed{frac{7}{8}}. ]

answer:1. **Convert the work shift into minutes**: Jenna's work shift is 10 hours long. Since there are 60 minutes in an hour, the total time of her work shift is: [ 10 times 60 = 600 text{ minutes} ] 2. **Calculate the total time spent in meetings**: - The duration of the first meeting is 30 minutes. - The second meeting, being twice as long as the first, lasts: [ 2 times 30 = 60 text{ minutes} ] - The third meeting, being half as long as the second, lasts: [ frac{60}{2} = 30 text{ minutes} ] - The total meeting duration is: [ 30 + 60 + 30 = 120 text{ minutes} ] 3. **Calculate the percentage of the workday spent in meetings**: - The fraction of the workday spent in meetings is: [ frac{120 text{ minutes}}{600 text{ minutes}} ] - Convert this fraction to a percentage: [ frac{120}{600} times 100% = 20% ] 4. **Conclusion**: - Jenna spends 20% of her workday in meetings. - Thus, the answer is 20%. The final answer is boxed{20%} or boxed{(C)}.