answer:First, apply the power rule of logarithms: [log_3 (x+5)^2 = 2 log_3 (x+5).] Next, use the change of base formula for log_{1/3}: [log_{1/3} (x - 1) = frac{log_3 (x - 1)}{log_3 (1/3)} = -log_3 (x - 1).] Combine these transformations into the equation: [2 log_3 (x+5) - log_3 (x-1) = 4.] To solve for x, convert it back into a single logarithm: [log_3 ((x+5)^2 / (x-1)) = 4.] Exponentiate both sides with a base of 3: [(x+5)^2 = 3^4(x-1) quad text{where} quad 3^4 = 81.] Expanding and simplified equation: [(x + 5)^2 = 81(x - 1),] [x^2 + 10x + 25 = 81x - 81,] [x^2 - 71x + 106 = 0.] Solving this quadratic equation using the quadratic formula: [x = frac{-(-71) pm sqrt{(-71)^2 - 4 cdot 1 cdot 106}}{2 cdot 1},] [x = frac{71 pm sqrt{5041 - 424}}{2},] [x = frac{71 pm sqrt{4617}}{2}.] The exact roots are: [x = frac{71 + sqrt{4617}}{2} quad text{or} quad x = frac{71 - sqrt{4617}}{2}.] However, as x-1 implies x>1, we take x = frac{71 + sqrt{4617}}{2} as the valid root. Thus, assuming sqrt{4617} simplifies correctly and remains consistent with the original equation, x = boxed{frac{71 + sqrt{4617}}{2}}.

answer:From the given conditions, we can derive that frac {1}{b}-3b=a+ frac {1}{a}. According to the basic inequality, we have frac {1}{b}-3b=a+ frac {1}{a}≥2 sqrt {acdot frac {1}{a}}=2. The equality holds if and only if a= frac {1}{a} (given that a>0), which is true when a=1. Therefore, frac {1}{b}-3b≥2. Since b>0, we have 3b^2+2b-1≤0. Solving this inequality, we get 0<b≤ frac {1}{3}. Hence, the maximum value of b is boxed{frac {1}{3}}. By using the given conditions, we derive the equation frac {1}{b}-3b=a+ frac {1}{a}. Then, applying the basic inequality, we find that frac {1}{b}-3b≥2. Solving this inequality and considering that b>0, we determine the range of b's possible values, and thus, find its maximum value. This problem tests the application of basic inequalities. The key to solving it is to use the basic inequality to find the range of the algebraic expression and the range of the parameter. It requires computational skills and is of moderate difficulty.

answer:To split the bill evenly among 8 people, we need to divide the total bill by the number of people. Total bill: 314.12 Number of people: 8 314.12 ÷ 8 = 39.265 Since 1 cent is the smallest unit, we cannot have a fraction of a cent. Therefore, we need to round to the nearest cent. The amount 39.265 would be rounded to 39.27 when considering the nearest cent. So, each person would pay boxed{39.27} .

answer:The longest segment inside a cylinder forms a right triangle with the cylinder's height and diameter as the triangle's legs. The diameter of the cylinder is twice the radius: text{Diameter} = 2 times 5 text{ cm} = 10 text{ cm}. The height of the cylinder is given as: text{Height} = 12 text{ cm}. Using the Pythagorean theorem, the length of the longest segment, or the hypotenuse (h), is calculated as: h = sqrt{(text{Diameter})^2 + (text{Height})^2} = sqrt{10^2 + 12^2} = sqrt{100 + 144} = sqrt{244}. Hence, the longest segment that fits inside the cylinder is (boxed{sqrt{244} text{ cm}}).