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question:the 5th century BC, the ancient Greek philosopher Zeno proposed the famous paradox of Achilles and the Tortoise: Suppose that Achilles races a tortoise, giving the tortoise a head start of 1,000 meters, and Achilles runs at a speed 10 times faster than the tortoise. When Achilles has run 1,000 meters, the tortoise has moved 100 meters ahead; when Achilles has run another 100 meters, the tortoise is still 10 meters ahead; when Achilles has run another 10 meters, the tortoise is still 1 meter ahead, and so on. Thus, Achilles can never catch up with the tortoise. According to this pattern, if the distance between Achilles and the tortoise is exactly 0.01 meters, the total distance the tortoise has traveled is ( ): A: frac {10^{4}1}{90} B: frac {10^{5}1}{900} C: frac {10^{5}9}{90} D: frac {10^{4}9}{900}
answer:From the problem, we know that the sequence of distances the tortoise travels forms a geometric series {a_n}, with a_1 = 100, q = frac{1}{10}, and a_n = 0.01. The sum S_n of this geometric series represents the total distance traveled by the tortoise and is given by the formula: S_n = frac{a_1 - a_nq}{1-q} = frac{100 - 0.01 times frac{1}{10}}{1 - frac{1}{10}} = boxed{frac{10^5-1}{900}}
question:The greatest common divisor (GCD) and the least common multiple (LCM) of 45 and 150 are ( ) A: 5,150 B: 15,450 C: 450,15 D: 15,150
answer:Use the Euclidean algorithm to find the GCD of 45 and 150: 150 = 45 times 3 + 15, and 45 = 15 times 3. Therefore, the GCD of 45 and 150 is 15. Thus, the LCM of 45 and 150 is 15 times (45 div 15) times (150 div 15) = 450. Therefore, the correct answer is boxed{text{B}}.
question:Given that ( f(x) ) and ( g(x) ) are both functions defined on ( mathbf{R} ), the graph of ( f(x) ) is symmetric about the line ( x=1 ), the graph of ( g(x) ) is symmetric about the point ( (1, 2) ), and ( f(x) + g(x) = 9^{x} + x^{3} + 1 ). Find the value of ( f(2) cdot g(2) ).
answer:Given the functions ( f(x) ) and ( g(x) ) which are defined on ( mathbb{R} ), the graph of ( f(x) ) is symmetrical about the line ( x = 1 ), and the graph of ( g(x) ) is symmetrical about the point ((1, -2)). Moreover, we are given that: [ f(x) + g(x) = 9^x + x^3 + 1 ] We need to find the value of ( f(2) cdot g(2) ). Let's follow the steps: 1. Due to the symmetry of ( f(x) ) about ( x = 1), we have: [ f(2) = f(0) ] 2. Due to the point symmetry of ( g(x) ) about ((1, -2)), we have: [ g(2) = -g(0) - 4 ] 3. We first evaluate the given equation at ( x = 0 ): [ f(0) + g(0) = 9^0 + 0^3 + 1 ] [ f(0) + g(0) = 1 + 1 ] [ f(0) + g(0) = 2 ] 4. Next, we evaluate the given equation at ( x = 2 ): [ f(2) + g(2) = 9^2 + 2^3 + 1 ] [ f(2) + g(2) = 81 + 8 + 1 ] [ f(2) + g(2) = 90 ] 5. Using the symmetry conditions and the results from steps 3 and 4, we have: [ f(2) = f(0) ] [ g(2) = -g(0) - 4 ] Substituting these into the expressions from steps 3 and 4, we get: [ f(0) + g(0) = 2 ] [ f(0) + (-g(0) - 4) = 90 ] 6. Let ( f(0) = a ) and ( g(0) = b ). From our equations, we have: [ a + b = 2 ] [ a - b - 4 = 90 ] From ( a - b - 4 = 90 ): [ a - b = 94 ] 7. Now we have the system of linear equations: [ a + b = 2 ] [ a - b = 94 ] Adding these two equations: [ (a + b) + (a - b) = 2 + 94 ] [ 2a = 96 ] [ a = 48 ] Substituting ( a = 48 ) back into ( a + b = 2 ): [ 48 + b = 2 ] [ b = 2 - 48 ] [ b = -46 ] 8. Thus, we find: [ f(2) = f(0) = a = 48 ] [ g(2) = -g(0) - 4 = -(-46) - 4 = 46 - 4 = 42 ] 9. Therefore, we can compute ( f(2) cdot g(2) ): [ f(2) cdot g(2) = 48 cdot 42 ] Calculating the product: [ 48 cdot 42 = (50 - 2)(40 + 2) ] [ = 50 cdot 40 + 50 cdot 2 - 2 cdot 40 - 2 cdot 2 ] [ = 2000 + 100 - 80 - 4 ] [ = 2000 + 100 - 80 - 4 ] [ = 2016 ] The value of ( f(2) cdot g(2) ) is (boxed{2016}).
question:A student's scores in four shooting tests (unit: rings) are 9, 9, x, and 8, respectively. If the mode of this data set is equal to the mean, then the value of x is ( ) A: 10 B: 9 C: 8 D: 7
answer:Given the problem, we are to find the value of x in a set of scores 9, 9, x, and 8 such that the mode equals the mean. 1. **Understanding the Mode**: Since the mode equals the mean, and we have two 9s, the mode must be 9. This is because for the mode to equal the mean, the most frequent score must also influence the average to match it. Given the scores, 9 is the only number that appears more than once initially, making it the mode. 2. **Calculating the Mean**: The mean (average) of the scores is calculated by summing all the scores and dividing by the number of scores. We have the scores 9, 9, x, and 8. The mean is also given to be 9, which is the same as the mode. 3. **Setting Up the Equation**: To find x, we set up the equation based on the mean formula: [ frac{9 + 9 + x + 8}{4} = 9 ] 4. **Solving for x**: Simplifying the equation, we multiply both sides by 4 to get rid of the denominator: [ 9 + 9 + x + 8 = 9 times 4 ] Simplifying further: [ 26 + x = 36 ] Subtracting 26 from both sides to solve for x: [ x = 36 - 26 ] [ x = 10 ] Therefore, the value of x that makes the mode equal to the mean in this data set is boxed{10}, which corresponds to choice A.