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āEvaluate lfloor e rfloor.
š”Okay, so I need to figure out the floor of e, which is written as ā£eā¦. Hmm, I remember that the floor function gives the greatest integer less than or equal to a given number. So, if I can find out what e is approximately, I can determine the floor value.First, I think e is Euler's number, right? Yeah, it's an important constant in mathematics. I recall that e is approximately 2.71828. Let me double-check that. Yeah, I think that's correct. So, e is roughly 2.71828.Now, applying the floor function to e means I need to find the largest integer that is not greater than e. Since e is about 2.71828, the integers around it are 2 and 3. But 3 is greater than e, so the floor can't be 3. Therefore, the floor of e must be 2.Wait, let me make sure I'm not missing anything. Is there a possibility that e could be exactly 3? No, because I remember e is an irrational number, meaning it can't be expressed as a simple fraction, and its decimal goes on forever without repeating. So, it's definitely not exactly 3. It's somewhere between 2 and 3, closer to 3 but not reaching it.Therefore, the greatest integer less than or equal to e is 2. I think that's solid. I don't see any mistakes in my reasoning. So, the floor of e is 2.
āThere are some stamps with denominations of 0.5 yuan, 0.8 yuan, and 1.2 yuan, with a total value of 60 yuan. The number of 0.8 yuan stamps is four times the number of 0.5 yuan stamps. How many 1.2 yuan stamps are there?
š”Alright, let's tackle this stamp problem. So, we have stamps of three denominations: 0.5 yuan, 0.8 yuan, and 1.2 yuan. The total value of all these stamps is 60 yuan. Also, it's given that the number of 0.8 yuan stamps is four times the number of 0.5 yuan stamps. We need to find out how many 1.2 yuan stamps there are.First, I'll try to define variables for the number of each type of stamp. Let's say:- Let ( x ) be the number of 0.5 yuan stamps.- Since the number of 0.8 yuan stamps is four times that of 0.5 yuan stamps, we can denote the number of 0.8 yuan stamps as ( 4x ).- Let ( y ) be the number of 1.2 yuan stamps.Now, the total value contributed by each type of stamp would be:- For 0.5 yuan stamps: ( 0.5x ) yuan.- For 0.8 yuan stamps: ( 0.8 times 4x = 3.2x ) yuan.- For 1.2 yuan stamps: ( 1.2y ) yuan.Adding these up should give the total value of 60 yuan. So, the equation becomes:[0.5x + 3.2x + 1.2y = 60]Combining like terms:[3.7x + 1.2y = 60]Hmm, dealing with decimals can be a bit messy, so maybe I can eliminate them by multiplying the entire equation by 10 to make the numbers whole:[37x + 12y = 600]Now, I need to solve for ( y ). Let's rearrange the equation:[12y = 600 - 37x][y = frac{600 - 37x}{12}]Since ( y ) must be an integer (you can't have a fraction of a stamp), the numerator ( 600 - 37x ) must be divisible by 12. So, ( 37x ) must leave the same remainder as 600 when divided by 12.Let's find out what 600 is modulo 12:[600 div 12 = 50 text{ with a remainder of } 0]So, ( 600 equiv 0 mod 12 ).Therefore, ( 37x ) must also be congruent to 0 modulo 12:[37x equiv 0 mod 12]But 37 divided by 12 is 3 with a remainder of 1, so:[37 equiv 1 mod 12]Thus,[37x equiv x mod 12]So,[x equiv 0 mod 12]This means ( x ) must be a multiple of 12. Let's try the smallest multiple, which is 12.Plugging ( x = 12 ) back into the equation for ( y ):[y = frac{600 - 37 times 12}{12} = frac{600 - 444}{12} = frac{156}{12} = 13]So, ( y = 13 ). That means there are 13 stamps of 1.2 yuan.Just to double-check, let's calculate the total value:- 0.5 yuan stamps: ( 12 times 0.5 = 6 ) yuan- 0.8 yuan stamps: ( 48 times 0.8 = 38.4 ) yuan- 1.2 yuan stamps: ( 13 times 1.2 = 15.6 ) yuanAdding them up: ( 6 + 38.4 + 15.6 = 60 ) yuan. Perfect, that matches the total value given.So, the number of 1.2 yuan stamps is 13.
āThe center of circle C is C(2,0), and it passes through point A(frac{3}{2},frac{sqrt{3}}{2}). (1) Find the standard equation of circle C. (2) Line l:kx+y+1=0 intersects circle C at points M and N, and |MN|=sqrt{2}. Find the value of k.
š”Okay, so I have this problem about a circle and a line intersecting it. Let me try to figure it out step by step. First, part (1) asks for the standard equation of circle C. I know that the standard equation of a circle is ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) is the center and (r) is the radius. The center is given as (C(2, 0)), so that part is straightforward. But I need to find the radius. The circle passes through point (Aleft(frac{3}{2}, frac{sqrt{3}}{2}right)). To find the radius, I can calculate the distance between the center (C(2, 0)) and point (A). The distance formula is (sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}). Plugging in the coordinates:Distance (r = sqrt{left(2 - frac{3}{2}right)^2 + left(0 - frac{sqrt{3}}{2}right)^2}).Let me compute that:First, (2 - frac{3}{2}) is (frac{1}{2}), and squaring that gives (frac{1}{4}).Next, (0 - frac{sqrt{3}}{2}) is (-frac{sqrt{3}}{2}), and squaring that gives (frac{3}{4}).Adding those together: (frac{1}{4} + frac{3}{4} = 1).So, the radius squared is 1, which means the radius (r) is 1. Therefore, the standard equation of the circle is ((x - 2)^2 + y^2 = 1). That seems straightforward.Now, moving on to part (2). We have a line (l: kx + y + 1 = 0) that intersects the circle at points (M) and (N), and the distance between (M) and (N) is (sqrt{2}). We need to find the value of (k).Hmm, okay. I remember that the length of the chord in a circle can be found using the formula (|MN| = 2sqrt{r^2 - d^2}), where (d) is the distance from the center of the circle to the line. Given that (|MN| = sqrt{2}) and (r = 1), let's plug these into the formula:(sqrt{2} = 2sqrt{1^2 - d^2}).Dividing both sides by 2:(frac{sqrt{2}}{2} = sqrt{1 - d^2}).Squaring both sides:(left(frac{sqrt{2}}{2}right)^2 = 1 - d^2).Calculating the left side:(frac{2}{4} = frac{1}{2}).So, (frac{1}{2} = 1 - d^2).Solving for (d^2):(d^2 = 1 - frac{1}{2} = frac{1}{2}).Therefore, (d = frac{sqrt{2}}{2}).Now, I need to find the distance from the center (C(2, 0)) to the line (kx + y + 1 = 0). The formula for the distance from a point ((x_0, y_0)) to the line (Ax + By + C = 0) is:(d = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}}).In our case, the line is (kx + y + 1 = 0), so (A = k), (B = 1), and (C = 1). The point is (C(2, 0)), so (x_0 = 2), (y_0 = 0).Plugging into the formula:(d = frac{|k cdot 2 + 1 cdot 0 + 1|}{sqrt{k^2 + 1^2}} = frac{|2k + 1|}{sqrt{k^2 + 1}}).We already found that (d = frac{sqrt{2}}{2}), so:(frac{|2k + 1|}{sqrt{k^2 + 1}} = frac{sqrt{2}}{2}).To solve for (k), let's square both sides to eliminate the square roots:(left(frac{|2k + 1|}{sqrt{k^2 + 1}}right)^2 = left(frac{sqrt{2}}{2}right)^2).Simplifying:(frac{(2k + 1)^2}{k^2 + 1} = frac{2}{4} = frac{1}{2}).Cross-multiplying:(2(2k + 1)^2 = (k^2 + 1)).Expanding the numerator:(2(4k^2 + 4k + 1) = k^2 + 1).Multiplying through:(8k^2 + 8k + 2 = k^2 + 1).Bringing all terms to one side:(8k^2 + 8k + 2 - k^2 - 1 = 0).Simplifying:(7k^2 + 8k + 1 = 0).Now, we have a quadratic equation: (7k^2 + 8k + 1 = 0). Let's solve for (k) using the quadratic formula:(k = frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a = 7), (b = 8), and (c = 1).Calculating the discriminant:(b^2 - 4ac = 64 - 28 = 36).So, the solutions are:(k = frac{-8 pm sqrt{36}}{14} = frac{-8 pm 6}{14}).This gives two solutions:1. (k = frac{-8 + 6}{14} = frac{-2}{14} = -frac{1}{7}).2. (k = frac{-8 - 6}{14} = frac{-14}{14} = -1).So, the possible values for (k) are (-1) and (-frac{1}{7}).Let me just double-check my steps to make sure I didn't make any mistakes. Starting from the distance formula, then setting up the equation, squaring both sides, expanding, and solving the quadratic. It seems correct. I think these are the right values for (k).**Final Answer**(1) The standard equation of circle (C) is (boxed{(x - 2)^2 + y^2 = 1}).(2) The value of (k) is (boxed{-1}) or (boxed{-dfrac{1}{7}}).
āWhat is the remainder when 121212 . . . . . . . (300 digits) is divided by 99?
š”Okay, so I have this problem where I need to find the remainder when a really long number, like 121212... with 300 digits, is divided by 99. Hmm, that seems a bit intimidating at first because the number is so long, but maybe there's a pattern or a trick I can use to make it simpler.First, I notice that the number is just the sequence "12" repeated over and over again. Since there are 300 digits, and each "12" is two digits, that means the sequence "12" is repeated 150 times. So, the number looks like 121212...12, with 150 "12"s.Now, I need to find the remainder when this number is divided by 99. Dividing such a large number directly seems impossible, so I should think about properties of numbers and modular arithmetic that might help.I remember that 99 is a multiple of 9 and 11, and there are some divisibility rules for these numbers. Maybe I can use those rules to find the remainder when divided by 99. But I'm not sure if that's the best approach here.Another thought: since 99 is 100 minus 1, maybe I can use that to simplify the problem. Let me think about how the number is constructed. Each "12" is like adding 12 multiplied by some power of 100. For example, the first "12" is 12, the next "12" is 12 times 100, the next is 12 times 100 squared, and so on, up to 12 times 100 to the power of 149 (since there are 150 terms).So, mathematically, the number can be written as:12 + 12*100 + 12*100^2 + ... + 12*100^149I can factor out the 12:12*(1 + 100 + 100^2 + ... + 100^149)Now, I need to find the remainder of this sum when divided by 99. Since 100 is congruent to 1 modulo 99 (because 100 - 99 = 1), each term 100^k is congruent to 1^k, which is 1, modulo 99.So, the sum inside the parentheses becomes:1 + 1 + 1 + ... + 1 (150 times)Which is just 150. Now, 150 divided by 99 is 1 with a remainder of 51 (since 99*1 = 99, and 150 - 99 = 51).So, the sum inside the parentheses is congruent to 51 modulo 99.Now, multiplying this by 12 gives:12*51 = 612But 612 is still larger than 99, so I need to find the remainder when 612 is divided by 99. Dividing 612 by 99 gives 6*99 = 594, and 612 - 594 = 18.So, the remainder is 18.Wait, let me double-check that. If I have 12 repeated 150 times, and each "12" is contributing 12*100^k, and since 100 ā” 1 mod 99, each term is just 12*1, so the total sum is 12*150 = 1800. Then, 1800 divided by 99: 99*18 = 1782, and 1800 - 1782 = 18. Yeah, that matches.So, the remainder is indeed 18.
āJasmine initially had 5 paperclips on Monday. Each subsequent day, her number of paperclips doubled. On what day of the week did she first have more than 200 paperclips?
š”First, identify the initial number of paperclips and the doubling pattern. Jasmine starts with 5 paperclips on Monday, and the number doubles each day.Next, set up the inequality to find when the number of paperclips exceeds 200:5 * 2^k > 200Solve for k:2^k > 40Test successive powers of 2 to find the smallest k that satisfies the inequality:- 2^5 = 32 (not greater than 40)- 2^6 = 64 (greater than 40)Determine the corresponding day of the week. Since k=0 is Monday, k=6 corresponds to Sunday.Therefore, Jasmine first has more than 200 paperclips on Sunday.
āEddy and Freddy start simultaneously from city A and they travel to city B and city C respectively. Eddy takes 3 hours and Freddy takes 4 hours to complete the journey. The distance between city A and city B is 600 km and city A and city C is 460 km. What is the ratio of their average speed of travel? (Eddy : Freddy)
š”First, I need to calculate the average speed for both Eddy and Freddy. Average speed is determined by dividing the total distance traveled by the total time taken.For Eddy traveling from city A to city B:- Distance = 600 km- Time = 3 hours- Average Speed = 600 km / 3 hours = 200 km/hFor Freddy traveling from city A to city C:- Distance = 460 km- Time = 4 hours- Average Speed = 460 km / 4 hours = 115 km/hNext, I'll determine the ratio of their average speeds by dividing Eddy's average speed by Freddy's average speed:- Ratio = 200 km/h / 115 km/hTo simplify the ratio, I'll divide both numbers by their greatest common divisor, which is 5:- Simplified Ratio = (200/5) km/h / (115/5) km/h = 40 km/h / 23 km/hTherefore, the ratio of their average speeds (Eddy : Freddy) is 40:23.