answer:To solve the problem, we need to determine the average waiting time for the first bite given that: - The first fishing rod averages 3 bites in 6 minutes. - The second fishing rod averages 2 bites in 6 minutes. - Together, they average 5 bites in 6 minutes. 1. **Calculate the average rate of bites per minute for both rods combined:** Given that both rods together average 5 bites in 6 minutes, we can calculate the average rate per minute: [ text{Average rate} = frac{text{Total Bites}}{text{Total Time}} = frac{5 text{ bites}}{6 text{ minutes}} = frac{5}{6} text{ bites per minute} ] 2. **Determine the average waiting time for a bite:** The average waiting time can be found as the reciprocal of the average rate of bites per minute: [ text{Average waiting time} = frac{1}{text{Average rate}} = frac{1}{frac{5}{6}} = frac{6}{5} = 1.2 text{ minutes} ] 3. **Convert the average waiting time into minutes and seconds:** We know: [ 1 text{ minute} + 0.2 text{ minutes} ] Convert (0.2) minutes to seconds: [ 0.2 text{ minutes} = 0.2 times 60 text{ seconds} = 12 text{ seconds} ] Thus, (1.2) minutes can be expressed as: [ 1 text{ minute} text{ and } 12 text{ seconds} ] # Conclusion: The average waiting time is 1 minute 12 seconds. [ boxed{1 text{ minute 12 seconds}} ]

answer:1. **Original Cube Characteristics**: - Original cube has side length 4, with 8 vertices, 12 edges, and 6 faces. 2. **Removing Diverse-Sized Cubes**: - Three corners are affected with different cube sizes removed: two corners remove (1 times 1 times 1) cubes and one removes a (2 times 2 times 2) cube. 3. **Edges Impact Analysis**: - Each (1 times 1 times 1) cube normally would have 12 edges, but only edges on the exposed surfaces come into new play. However, cube overlaps severely reduce net new edges because corners are shared. - As analyzed originally, removing a cube affects edges radically. Remember that each removed (1 times 1 times 1) corner removes a full set of edges for each face on the surface, simplifying to not adding new edges in total structure. - The (2 times 2 times 2) cube significantly alters the edge count. Each of the three faces exposed at the removed corner adds 16 new edges (excluding shared edges, which would be debated in shared sides but accepting covered four at each mating side). 4. **Calculating Total Edges**: - Original cube has 12 edges. - Subtract three edges per removed (1 times 1 times 1) cube directly impacting the original edges (removing 6 directly impacted edges). - The (2 times 2 times 2) cube adds 16 new edges, after adjusting shared edges. - Net new edges in structure: 12 - 6 + 16 = 22 edges. Thus, considering alterations and asymmetry, the total number of edges becomes 22. The final answer is boxed{textbf{(C) }22}.

answer:The steps to solve the problem are: 1. **Calculation of the number of ways to choose which three of the seven picks are blue**: Using the binomial coefficient, calculate binom{7}{3}. 2. **Probability of picking a blue marble**: Since there are 8 blue marbles out of a total of 15, the probability of picking a blue marble each time is frac{8}{15}. 3. **Probability of picking a red marble**: Since there are 7 red marbles out of a total of 15, the probability of picking a red marble each time is frac{7}{15}. 4. **Calculating the probability that exact three out of seven picks are blue**: Multiply the number of ways to arrange the blue and red marbles by the probability that each arrangement occurs. Using these steps: - Compute binom{7}{3} = 35. - The probability for a specific sequence where exactly three picks are blue and the remaining four are red is left(frac{8}{15}right)^3 left(frac{7}{15}right)^4. - Thus, the probability of exactly three blue picks in seven tries is 35 times left(frac{8}{15}right)^3 left(frac{7}{15}right)^4 = 35 times frac{512}{3375} times frac{2401}{50625} = frac{37800320}{91293225}. Verify and simplify the final probability: [ frac{37800320}{91293225} = frac{640}{1547} ] So, the solution is: [ boxed{frac{640}{1547}} ]

answer:- **Perimeter calculation**: In a regular decagon, all sides are of equal length. - There are 10 sides in a decagon. - Each side has a length of 3. To find the perimeter (P) of the decagon, use the formula: [ P = text{number of sides} times text{length of each side} ] [ P = 10 times 3 = 30 ] Thus, the perimeter of the decagon is (boxed{30}) units.