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question:Points A, B, and D are located in a plane where B is at (1,1) and D is at (4,5). Point A is on the xaxis at (16,0). Calculate the sum of the distances AD and BD. A) 16 B) 17 C) 18 D) 19 E) 20
answer:1. **Calculate BD using Pythagoras' Theorem:** - BD = sqrt{(4 - 1)^2 + (5 - 1)^2} = sqrt{3^2 + 4^2} = sqrt{9 + 16} = sqrt{25} = 5. 2. **Calculate AD using Pythagoras' Theorem:** - The horizontal distance between A and D is 16 - 4 = 12, and the vertical distance is 5 - 0 = 5. - AD = sqrt{(16 - 4)^2 + (5 - 0)^2} = sqrt{12^2 + 5^2} = sqrt{144 + 25} = sqrt{169} = 13. 3. **Calculate the sum AD + BD:** - AD + BD = 13 + 5 = 18. Conclusion: The sum of the distances AD and BD is 18. The final answer is boxed{C}.
question:Given f(3^x) = 2xlog_{2}3, then f(2) = ?
answer:**Method 1:** Let 3^x = t, then we get x = log_{3}t. Substituting this into the original function, we obtain f(t) = 2log_{3}t times log_{2}3 = 2log_{2}t. Therefore, f(2) = 2log_{2}2 = 2. Hence, the answer is boxed{2}. **Method 2:** Let 3^x = 2, then we get x = log_{3}2. Substituting this into the original function, we obtain f(2) = 2log_{3}2 times log_{2}3 = 2. Hence, the answer is boxed{2}.
question:First a is chosen at random from the set {1,2,3,cdots,99,100}, and then b is chosen at random from the same set. The probability that the integer 2^a + 5^b has units digit 4 is_______? A) frac{1}{16} B) frac{1}{8} C) frac{3}{16} D) frac{1}{5} E) 0
answer:1. **Cyclic Nature of Units Digits for Powers of 2 and 5**: - Powers of 2: 2, 4, 8, 6 repeat every 4 numbers. - Powers of 5: 5, 5, 5, 5 remain constant at 5. 2. **Calculate the Length of the Cycle**: - The unit digit of 2^a cycles with a length of 4 and 5^b has a constant cycle of 4. 3. **Possible Units Digit Patterns**: - Pattern count for each 2^a (a in {2, 4, 8, 6}) when added to 5 for each cycle: - 2+5 = 7 - 4+5 = 9 - 8+5 = 13 → 3 - 6+5 = 11 → 1 4. **Identify Favorable Outcomes Where Sum Has Units Digit 4**: - From the patterns above, only 6+5 = 11 results in a units digit of 1. However, we want the sum to have a units digit of 4, which doesn't appear. 5. **Conclusion**: - No combinations of 2^a and 5^b yield a sum with a units digit of 4, therefore the probability is 0. The final answer is E) boxed{0}
question:Convert 35<sub>(10)</sub> to binary.
answer:Since 35 div 2 = 17 remainder 1, 17 div 2 = 8 remainder 1, 8 div 2 = 4 remainder 0, 4 div 2 = 2 remainder 0, 2 div 2 = 1 remainder 0, 1 div 2 = 0 remainder 1, Therefore, (35)_{10} = (100011)_{2}, So, the answer is: (100011)_{2} By using the division by 2 and taking the remainder method, we can convert the decimal number 35 into a binary number. This problem tests the knowledge of converting between decimal and other number systems, where mastering the conversion between decimal and binary using the division by 2 and taking the remainder method is a basic knowledge check. boxed{(100011)_2}