answer:1. **Assessment of Error and Reverse Engineering**: Ron's operation was (text{reversed } a) cdot b + 5 = 266. 2. **Solve for Original Product**: [(text{reversed } a) cdot b = 266 - 5 = 261] 3. **Prime Factorization of 261**: [261 = 3 times 3 times 29 = 9 times 29] This suggests possible values for a after reversing and b. 4. **Determine Possible Values for a and b**: - If the reversed a = 9, then a = 90, which is not valid as a two-digit number. - If the reversed a = 29, then a = 92, which is valid. 5. **Confirming Match with b = 9**: - Reversing 92 gives 29 and multiplying by 9 yields 261, which after adding 5, matches Ron's erroneous result of 266. 6. **Calculate the Correct Product**: [ a times b = 92 times 9 = 828 ] 7. **Conclusion**: The correct value of the product of a and b is 828. The final answer is boxed{828 textbf{(B)}}

answer:**Analysis** This problem tests the understanding of the proof by contradiction method, which is quite basic. Assume frac{b}{a} < 0 and frac{c}{a} < 0. According to the basic inequality, we have frac{b}{a} + frac{c}{a} < -2, which contradicts frac{b}{a} + frac{c}{a} geqslant -2. Therefore, frac{b}{a} > 0 and frac{c}{a} > 0, indicating that a, b, and c have the same sign. **Solution Steps** 1. From frac{b}{a} cdot frac{c}{a} > 1, we know that frac{b}{a} and frac{c}{a} have the same sign. 2. Consider the case when frac{b}{a} > 0 and frac{c}{a} > 0. The inequality frac{b}{a} + frac{c}{a} geqslant -2 obviously holds. 3. Now, let's consider the case when frac{b}{a} < 0 and frac{c}{a} < 0. This leads to - frac{b}{a} > 0 and - frac{c}{a} > 0. 4. Adding these two inequalities together, we get left(- frac{b}{a}right) + left(- frac{c}{a}right) geqslant 2. 5. Taking the square root of both sides, we obtain sqrt{left(- frac{b}{a}right) cdot left(- frac{c}{a}right)} > 2, which results in frac{b}{a} + frac{c}{a} < -2. 6. This contradicts the given condition frac{b}{a} + frac{c}{a} geqslant -2. Thus, our assumption that frac{b}{a} < 0 and frac{c}{a} < 0 is incorrect. 7. Consequently, we conclude that frac{b}{a} > 0 and frac{c}{a} > 0, meaning that a, b, and c have the same sign. Therefore, the correct answer is boxed{text{A}}.

answer:Let's denote the width of the cistern as ( w ) meters. The wet surface of the cistern includes the bottom, the two longer sides, and the two shorter sides (the water surface is not included since it's not a "wet" surface). We can calculate the area of each of these surfaces: 1. The bottom surface area is ( 12 times w ) square meters. 2. The area of each of the longer sides is ( 1.25 times 12 ) square meters (since the depth is 1.25 meters). 3. The area of each of the shorter sides is ( 1.25 times w ) square meters. The total wet surface area is the sum of these areas: [ text{Total wet surface area} = text{Bottom area} + 2 times text{Longer side area} + 2 times text{Shorter side area} ] [ 88 = (12 times w) + 2 times (1.25 times 12) + 2 times (1.25 times w) ] [ 88 = 12w + 2 times 15 + 2.5w ] [ 88 = 12w + 30 + 2.5w ] [ 88 = 14.5w + 30 ] Now, let's solve for ( w ): [ 88 - 30 = 14.5w ] [ 58 = 14.5w ] [ w = frac{58}{14.5} ] [ w = 4 ] So, the width of the cistern is boxed{4} meters.

answer:Since z=3+4i, we have |z|=5, Therefore, frac {z}{|z|}=frac {3+4i}{5}= frac {3}{5}+ frac {4}{5}i. Hence, the correct option is: boxed{C}. By calculating |z| and substituting it into frac {z}{|z|}, we obtain the answer. This question tests the operation of multiplication and division in the algebraic form of complex numbers, focusing on how to calculate the modulus of a complex number, and is considered a basic question.