answer：Okay, so Sue's family went on vacation, and her mom was driving the car at 60 miles per hour. They camped at a campground after traveling for 5 hours. I need to figure out how far the campground is from their home. Hmm, let's see. First, I remember that distance is calculated by multiplying speed by time. So, if the speed is 60 mph and the time is 5 hours, I can find the distance by doing 60 times 5. Let me write that down: 60 mph multiplied by 5 hours. Wait, does that make sense? Yeah, because if you're going 60 miles every hour, then in one hour you cover 60 miles, in two hours 120 miles, and so on. So, for 5 hours, it should be 60 times 5. Let me do the multiplication. 60 times 5 is 300. So, the distance should be 300 miles. Is there anything else I need to consider? Maybe check if the units make sense. Speed is in miles per hour, and time is in hours, so when I multiply them, the hours cancel out, leaving me with miles. That checks out. I don't think there are any other factors mentioned, like stops or different speeds. It just says her mom drove at 60 mph for 5 hours. So, I think 300 miles is the right answer. Let me double-check my calculation. 60 times 5 is indeed 300. Yep, that seems correct. So, the campground is 300 miles away from their home.

answer：Alright, so I've got this problem here about Billy, John, their mom, and their cousin Amy riding their bikes a certain number of times. I need to figure out the total number of times they all rode their bikes combined. Let me try to break this down step by step.First, it says Billy rode his bike 17 times. That seems straightforward. So, Billy's total rides are 17.Next, John rode his bike twice as many times as Billy. Okay, so if Billy rode 17 times, then John rode 2 times 17. Let me write that out: 2 * 17 = 34. So, John rode his bike 34 times.Now, their mother rode her bike 10 times more than John. Hmm, "10 times more" could be interpreted in a couple of ways. Does it mean 10 times the number of John's rides, or does it mean 10 rides added to John's total? The wording says "10 times more," which usually means multiplication, so I think it means 10 times John's rides. So, that would be 10 * 34. Let me calculate that: 10 * 34 = 340. So, their mother rode her bike 340 times.Then, their cousin Amy rode her bike 3 times more than the square root of the sum of John's and Billy's rides. Okay, this is a bit more complex. Let's break it down. First, I need to find the sum of John's and Billy's rides. John rode 34 times and Billy rode 17 times, so 34 + 17 = 51. Now, I need to find the square root of that sum. The square root of 51 is approximately 7.1414. Then, Amy rode her bike 3 times more than that square root. So, 3 * 7.1414 ≈ 21.4242. Since you can't ride a bike a fraction of a time, I think we need to round this to the nearest whole number, which would be 21.Now, to find the total number of rides, I need to add up all the rides from Billy, John, their mother, and Amy. So, Billy's 17, John's 34, their mother's 340, and Amy's 21. Let's add them up: 17 + 34 = 51, 51 + 340 = 391, and 391 + 21 = 412.Wait, that seems like a lot. Let me double-check my calculations to make sure I didn't make any mistakes. Billy rode 17 times, John rode twice as much, which is 34. Their mother rode 10 times more than John, so 10 * 34 = 340. The sum of John's and Billy's rides is 34 + 17 = 51, and the square root of 51 is approximately 7.1414. Then, Amy rode 3 times that, which is about 21.4242, rounded to 21. Adding them all up: 17 + 34 + 340 + 21 = 412.Hmm, that still seems high, especially the mother riding 340 times. Maybe I misinterpreted "10 times more." If "10 times more" means adding 10 times John's rides to his original count, then it would be 34 + (10 * 34) = 34 + 340 = 374. But that would make the mother's rides even higher, which doesn't seem right. Alternatively, if "10 times more" just means 10 times John's rides, then it's 340, which is what I initially thought.Given that, I think the total is indeed 412 rides. But let me see if there's another way to interpret the problem that might result in a smaller total. Maybe "10 times more" means 10 additional rides on top of John's 34, making it 34 + 10 = 44. That would make the mother's rides 44. Then, Amy's rides would be 3 times the square root of (34 + 17) = 3 * sqrt(51) ≈ 21.4242, which rounds to 21. Adding them up: 17 + 34 + 44 + 21 = 116. That seems more reasonable.Wait, so depending on how "10 times more" is interpreted, the total could be either 412 or 116. That's a big difference. I need to clarify what "10 times more" means. In common language, "10 times more" usually means multiplying by 10, so 10 * John's rides = 340. But sometimes, people use "times more" to mean adding that multiple to the original amount, so 34 + (10 * 34) = 374. However, in mathematical terms, "10 times more" typically means 10 * the original amount, not adding it to the original.Given that, I think the correct interpretation is that the mother rode 10 times John's rides, which is 340. Therefore, the total rides would be 17 + 34 + 340 + 21 = 412.But I'm still unsure because the other interpretation leads to a much smaller and perhaps more plausible total of 116. Maybe the problem expects the second interpretation, where "10 times more" means adding 10 rides to John's total, making it 34 + 10 = 44. Then, Amy's rides would be 3 * sqrt(51) ≈ 21.4242, rounded to 21. So, total rides would be 17 + 34 + 44 + 21 = 116.To resolve this ambiguity, I think it's safer to go with the interpretation that "10 times more" means 10 times the original amount, so 10 * 34 = 340. Therefore, the total number of rides is 17 + 34 + 340 + 21 = 412.However, if the problem intends "10 times more" to mean adding 10 times the original amount to the original, making it 34 + 340 = 374, then the total would be even higher: 17 + 34 + 374 + 21 = 446.But that seems even more extreme. Given the context, it's more likely that "10 times more" means 10 times the original amount, so 340 rides for the mother. Therefore, the total is 412.But I'm still not entirely confident. Maybe I should look for similar problems or definitions to confirm the meaning of "10 times more." Upon checking, in mathematics, "times more" generally means multiplication by that factor. So, "10 times more" means 10 * the original number. Therefore, the mother rode 10 * 34 = 340 times.Given that, the total number of rides is 17 (Billy) + 34 (John) + 340 (Mother) + 21 (Amy) = 412.But to be thorough, let's consider both interpretations:1. "10 times more" = 10 * John's rides = 340 Total rides = 17 + 34 + 340 + 21 = 4122. "10 times more" = John's rides + 10 = 34 + 10 = 44 Total rides = 17 + 34 + 44 + 21 = 116Given the mathematical definition, the first interpretation is correct, leading to a total of 412 rides.However, in everyday language, people sometimes use "times more" to mean adding that multiple to the original amount. So, "10 times more" could mean 34 + 10*34 = 374. But that would make the mother's rides 374, which seems excessive.Alternatively, "10 times more" could mean 10 additional rides, making it 34 + 10 = 44. That would make the mother's rides 44, which seems more reasonable.Given the ambiguity, it's possible that the problem expects the second interpretation, leading to a total of 116 rides.To resolve this, I think it's best to present both interpretations and their respective totals, but based on mathematical definitions, the total should be 412.

answer：Okay, so I have this problem where I need to find the sum of B(n) from n=1 to n=255. B(n) is defined as the number of 1-runs in the binary representation of n. A 1-run is a non-extendable substring of 1s. For example, B(107) is 3 because 107 in binary is 1101011, which has three separate runs of 1s.First, I need to understand what a 1-run is. So, if I have a binary number like 1101011, the runs of 1s are the sequences of 1s that are not interrupted by 0s. In this case, the runs are at the beginning (11), then after the 0 (1), and then at the end (11). So that's three runs.Now, I need to calculate the sum of B(n) for all n from 1 to 255. That's a lot of numbers, so I need a smarter way than just converting each number to binary and counting the runs one by one.I remember that binary numbers from 1 to 255 are all the 8-bit numbers except for 0. So, each number can be represented as an 8-bit binary number, possibly with leading zeros. For example, 1 is 00000001, and 255 is 11111111.Maybe I can think about how often a new 1-run starts in each bit position across all numbers. If I can count the number of times a new 1-run starts in each position, then summing those counts should give me the total number of 1-runs.In an 8-bit number, the first bit is the most significant bit. If the first bit is 1, that's the start of a 1-run. Then, for each subsequent bit, if the current bit is 1 and the previous bit is 0, that's the start of a new 1-run.So, for each bit position from 1 to 8, I can calculate how many times a new 1-run starts in that position across all numbers from 1 to 255.Let's break it down:1. **First bit (most significant bit):** This bit is 1 for all numbers from 128 to 255. So, there are 128 numbers where the first bit is 1, each contributing one 1-run. So, the first bit contributes 128 runs.2. **Second bit:** The second bit alternates between 0 and 1 every 64 numbers. For example, from 0 to 63, the second bit is 0; from 64 to 127, it's 1; and so on. However, since we're starting from 1, the second bit is 0 for numbers 1 to 63, 1 for 64 to 127, 0 for 128 to 191, and 1 for 192 to 255.But wait, the second bit being 1 doesn't necessarily mean it's the start of a new run. It depends on the previous bit. If the first bit is 0 and the second bit is 1, that's a new run. But in our case, the first bit is 1 for numbers 128 to 255, so the second bit being 1 in those numbers doesn't start a new run because the first bit is already 1.Hmm, this complicates things. Maybe I need a different approach.Another idea: For each bit position from 2 to 8, the number of times a new 1-run starts at that position is equal to the number of times the current bit is 1 and the previous bit is 0 across all numbers.So, for each bit position i (from 2 to 8), the number of new runs starting at position i is equal to the number of numbers where bit i is 1 and bit i-1 is 0.Since each bit position is independent, except for the dependency on the previous bit, I can calculate this for each position.For position 2: The number of numbers where bit 2 is 1 and bit 1 is 0.Bit 1 is the most significant bit. It's 1 for numbers 128 to 255. So, if bit 1 is 1, bit 2 can be 0 or 1. But we need bit 2 to be 1 and bit 1 to be 0. However, bit 1 is 1 for numbers 128 to 255, so bit 1 is 0 for numbers 1 to 127.So, for numbers 1 to 127, bit 1 is 0. In these numbers, bit 2 can be 0 or 1. The number of numbers where bit 2 is 1 and bit 1 is 0 is 64 (since bit 2 is 1 for half of the numbers from 0 to 127, which is 64 numbers). But since we're starting from 1, it's still 64 numbers.Wait, but numbers from 1 to 127 include numbers where bit 1 is 0. So, for bit 2, in the range 1 to 127, bit 2 is 1 for 64 numbers (64 to 127). But in these numbers, bit 1 is 0 for numbers 1 to 127. Wait, no, bit 1 is the most significant bit, so for numbers 1 to 127, bit 1 is 0, and bit 2 can be 0 or 1.So, in numbers 1 to 127, bit 2 is 1 for 64 numbers (64 to 127). But in these numbers, bit 1 is 0, so each of these 64 numbers has a new 1-run starting at bit 2.Similarly, for numbers 128 to 255, bit 1 is 1, so if bit 2 is 1, it doesn't start a new run because bit 1 is already 1. So, the number of new runs starting at bit 2 is 64.Similarly, for bit 3: The number of numbers where bit 3 is 1 and bit 2 is 0.Bit 2 is 0 in half of the numbers. For each bit position, the number of times a bit is 0 is equal to the number of times it's 1, which is 128 for each bit in the full range 0 to 255. But since we're starting from 1, it's slightly different, but for large n, it's approximately half.Wait, maybe I can generalize this.For each bit position i (from 2 to 8), the number of times a new 1-run starts at position i is equal to the number of numbers where bit i is 1 and bit i-1 is 0.Since each bit is independent except for the dependency on the previous bit, the number of such numbers is 2^(8 - i). Because for each bit i, the previous bit i-1 must be 0, and the remaining bits can be anything.Wait, let's test this.For bit 2: The number of numbers where bit 2 is 1 and bit 1 is 0 is 2^(8 - 2) = 64. Which matches our earlier calculation.For bit 3: The number of numbers where bit 3 is 1 and bit 2 is 0 is 2^(8 - 3) = 32.Similarly, for bit 4: 16, bit 5: 8, bit 6: 4, bit 7: 2, bit 8: 1.So, the total number of new runs starting at each bit position from 2 to 8 is 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127.Adding the runs from the first bit, which is 128, the total number of runs is 128 + 127 = 255.Wait, that's interesting. So, the total sum S is 255.But let me double-check this.Each bit position from 1 to 8 contributes a certain number of runs. The first bit contributes 128 runs because it's 1 for all numbers from 128 to 255. Then, for each subsequent bit, the number of new runs is 2^(8 - i), where i is the bit position.So, for bit 2: 64, bit 3: 32, ..., bit 8: 1. Summing these gives 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127.Adding the 128 from the first bit, we get 255.Yes, that makes sense. So, the total sum of B(n) from n=1 to n=255 is 255.I think that's the answer.

answer：Okay, so I have this triangle ABC, and the sides opposite angles A, B, and C are given as a=3, b=√2, and c=2√2. I need to find the cosine of the largest internal angle of this triangle. Hmm, let me think about how to approach this.First, I remember that in any triangle, the largest angle is opposite the longest side. So, I should figure out which side is the longest. Let me list the sides: a=3, b=√2, and c=2√2. Calculating their approximate values might help. √2 is about 1.414, so b is approximately 1.414, and c is 2 times that, which is about 2.828. Comparing these to a, which is 3, it looks like a is the longest side. So, the largest angle should be opposite side a, which is angle A.Alright, so I need to find the cosine of angle A. I recall the Law of Cosines, which relates the sides of a triangle to the cosine of one of its angles. The formula is:cos A = (b² + c² - a²) / (2bc)Let me plug in the values I have into this formula. First, I'll compute each part step by step.Calculating b squared: b² = (√2)² = 2.Calculating c squared: c² = (2√2)² = (2)² * (√2)² = 4 * 2 = 8.Calculating a squared: a² = 3² = 9.Now, substituting these into the formula:cos A = (2 + 8 - 9) / (2 * √2 * 2√2)Let me simplify the numerator first: 2 + 8 is 10, and 10 - 9 is 1. So the numerator is 1.Now, the denominator: 2 * √2 * 2√2. Let me compute that step by step. First, 2 * 2 is 4. Then, √2 * √2 is 2. So, 4 * 2 is 8. Therefore, the denominator is 8.Putting it all together: cos A = 1 / 8.Wait, so the cosine of the largest angle, which is angle A, is 1/8. Let me double-check my calculations to make sure I didn't make a mistake.Starting again, b squared is (√2)^2 = 2, correct. c squared is (2√2)^2 = 8, that's right. a squared is 9, correct. So, numerator is 2 + 8 - 9 = 1, that's correct. Denominator: 2 * √2 * 2√2. Let me compute that again. 2 times 2 is 4, and √2 times √2 is 2, so 4 times 2 is 8. So, denominator is 8. So, 1 divided by 8 is indeed 1/8.Hmm, so that seems correct. Therefore, the cosine of the largest internal angle is 1/8. Looking at the options given: A is 1/8, B is 1/4, C is 3/8, and D is 1/2. So, the answer should be option A.Just to make sure, let me think if there's another way to approach this problem. Maybe using the Law of Sines? But the Law of Sines relates the sides and angles through sine, which might not be as straightforward for finding cosine. Plus, since we're dealing with the largest angle, and we know the sides, the Law of Cosines seems the most direct method.Alternatively, could I have confused the sides and angles? Let me confirm: side a is opposite angle A, which is correct. Since a is the longest side, angle A is the largest angle. So, yes, that makes sense.Another thought: maybe I should check if the triangle is valid. The sum of any two sides should be greater than the third side. Let's see: a=3, b≈1.414, c≈2.828.Check a + b > c: 3 + 1.414 ≈ 4.414 > 2.828, which is true.Check a + c > b: 3 + 2.828 ≈ 5.828 > 1.414, true.Check b + c > a: 1.414 + 2.828 ≈ 4.242 > 3, which is also true.So, the triangle inequality holds, meaning this is a valid triangle. Good, so no issues there.Therefore, I feel confident that my calculation is correct. The cosine of the largest internal angle is 1/8, which corresponds to option A.**Final Answer**boxed{A}

answer：Okay, so I have this cricket problem to solve, and I need to figure out how many runs a batsman should score in his 18th innings to increase his average by an additional 2 runs while also maintaining a personal scoring rate higher than the required run rate throughout his innings. Hmm, that sounds a bit complicated, but let's break it down step by step.First, let's understand the problem. The batsman has played 17 innings so far. In his 17th innings, he scored 85 runs, which increased his average by 3 runs. Now, in the 18th innings, the team is chasing a target of 150 runs in 20 overs, which means the required run rate is 7.5 runs per over. The batsman wants to increase his average by another 2 runs in this 18th innings while ensuring that his personal scoring rate is higher than 7.5 runs per over throughout his innings.Alright, so I need to find out how many runs he needs to score in the 18th innings to achieve this. Let's start by figuring out his average before the 17th innings.Let’s denote his average before the 17th innings as A. That means his total runs in 16 innings would be 16A. After scoring 85 runs in the 17th innings, his average increases by 3, so his new average becomes A + 3. The total runs after 17 innings would then be 16A + 85.But we can also express the total runs after 17 innings in terms of the new average: 17(A + 3). So, setting these equal:16A + 85 = 17(A + 3)Let me solve this equation for A.16A + 85 = 17A + 51Subtracting 16A from both sides:85 = A + 51Subtracting 51 from both sides:A = 34So, his average before the 17th innings was 34 runs. That means his total runs in 16 innings were 16 * 34 = 544 runs.After scoring 85 runs in the 17th innings, his total runs become 544 + 85 = 629 runs. His new average after 17 innings is 629 / 17 = 37 runs.Now, he wants to increase his average by an additional 2 runs after the 18th innings. So, his target average is 37 + 2 = 39 runs.To find out the total runs needed to achieve this average after 18 innings, we calculate 39 * 18 = 702 runs.He already has 629 runs after 17 innings, so he needs to score 702 - 629 = 73 runs in the 18th innings.But wait, there's another condition: he needs to maintain a personal scoring rate higher than the required run rate of 7.5 runs per over throughout his innings. So, he needs to score these 73 runs at a rate higher than 7.5 runs per over.Let me calculate how many overs he needs to score 73 runs at a rate higher than 7.5 runs per over.If he scores at exactly 7.5 runs per over, the number of overs required would be 73 / 7.5 ≈ 9.73 overs.But since he needs to maintain a higher rate, he should aim to score 73 runs in fewer than 9.73 overs. Since you can't play a fraction of an over in cricket, he needs to score 73 runs in 9 overs or less to maintain a higher scoring rate than 7.5 runs per over.Wait, but in the 18th innings, the team is chasing 150 runs in 20 overs, so the required run rate is 7.5 runs per over. The batsman needs to score 73 runs at a higher rate than 7.5 runs per over. So, he needs to score 73 runs in fewer than 9.73 overs, which is approximately 9 overs.But, hold on, the team is chasing 150 runs in 20 overs, so the required run rate is 7.5 runs per over. The batsman's personal scoring rate needs to be higher than this. So, if he scores 73 runs in, say, 9 overs, his personal run rate would be 73 / 9 ≈ 8.11 runs per over, which is higher than 7.5.Alternatively, if he scores 73 runs in 10 overs, his personal run rate would be 7.3 runs per over, which is lower than 7.5, so that wouldn't satisfy the condition.Therefore, he needs to score 73 runs in 9 overs or fewer to maintain a personal scoring rate higher than 7.5 runs per over.But wait, in cricket, an over consists of 6 balls, so 9 overs would be 54 balls. If he scores 73 runs in 54 balls, his scoring rate would be 73 / 9 ≈ 8.11 runs per over, which is indeed higher than 7.5.Alternatively, if he scores 73 runs in 8 overs, that would be 73 / 8 ≈ 9.125 runs per over, which is even higher.So, to satisfy both conditions: increasing his average by 2 runs and maintaining a personal scoring rate higher than 7.5 runs per over, he needs to score at least 73 runs in 9 overs or fewer.But, let me double-check my calculations to make sure I didn't make any mistakes.First, his average before the 17th innings was 34, total runs 544.After 17th innings: 544 + 85 = 629, average 37.Target average after 18th innings: 39.Total runs needed: 39 * 18 = 702.Runs needed in 18th innings: 702 - 629 = 73.Personal scoring rate: 73 runs in fewer than 9.73 overs, so 9 overs or fewer.Yes, that seems correct.But, just to be thorough, let's consider if he scores exactly 73 runs in 9 overs, his personal run rate is 73 / 9 ≈ 8.11 runs per over, which is higher than 7.5.If he scores 73 runs in 8 overs, his personal run rate is 73 / 8 ≈ 9.125 runs per over, which is also higher than 7.5.Therefore, as long as he scores 73 runs in 9 overs or fewer, he will meet both conditions.But, is there a possibility that scoring more than 73 runs could also work? Let's see.If he scores more than 73 runs, say 74 runs, then his average would increase by more than 2 runs. Specifically, the total runs would be 629 + 74 = 703, and the average would be 703 / 18 ≈ 39.06, which is an increase of approximately 2.06 runs, which is more than 2 runs.But the problem specifies increasing his average by an additional 2 runs, not more. So, scoring exactly 73 runs would increase his average by exactly 2 runs, from 37 to 39.Therefore, he needs to score exactly 73 runs in the 18th innings, and do so in 9 overs or fewer to maintain a personal scoring rate higher than 7.5 runs per over.Wait, but in cricket, you can't score a fraction of a run, so 73 runs is a whole number, which is fine.Also, in terms of balls, 9 overs are 54 balls, so he needs to score 73 runs in 54 balls or fewer.But, in practice, batsmen don't always score runs in whole numbers per over, but for the sake of this problem, we can consider the average runs per over.So, to sum up, the batsman needs to score 73 runs in his 18th innings, and he needs to do so in 9 overs or fewer to maintain a personal scoring rate higher than 7.5 runs per over.I think that's the solution. But let me just go through it one more time to make sure I didn't miss anything.1. Calculate average before 17th innings: A = 34.2. Total runs after 17 innings: 629.3. Target average after 18 innings: 39.4. Total runs needed: 702.5. Runs needed in 18th innings: 73.6. Personal scoring rate: 73 runs in 9 overs or fewer.Yes, that all checks out. So, the batsman needs to score 73 runs in his 18th innings, and do so in 9 overs or fewer to maintain a higher scoring rate than the required run rate.I think that's the answer.

answer：First, I need to determine Nancy's hourly rate by dividing her total earnings by the number of hours she worked.Next, with her hourly rate known, I can calculate the target amount she wants to earn by multiplying her hourly rate by the number of hours she plans to work.Finally, by performing these calculations, I will find out the exact target amount Nancy aims to achieve.