💡Okay, so I have this problem here: "Given that the surface area of a sphere is 16π, find the volume of the sphere." Hmm, alright, let's see. I remember that spheres have formulas for both surface area and volume, but I need to recall exactly what they are. First, surface area. I think the formula for the surface area of a sphere is something like 4πr², right? Yeah, that sounds familiar. So if the surface area is given as 16π, I can set up an equation using that formula to find the radius. Let me write that down:Surface Area = 4πr² = 16πOkay, so I need to solve for r. Let's see, if I divide both sides of the equation by π, that should cancel out π, leaving me with:4r² = 16Now, to solve for r², I can divide both sides by 4:r² = 16 / 4r² = 4Alright, so r squared is 4. To find r, I take the square root of both sides:r = √4r = 2Okay, so the radius of the sphere is 2 units. Now that I have the radius, I can find the volume. I remember the formula for the volume of a sphere is (4/3)πr³. Let me write that down:Volume = (4/3)πr³Since I found that r = 2, I can substitute that into the formula:Volume = (4/3)π(2)³Now, I need to calculate (2)³. That's 2 * 2 * 2, which is 8. So now I have:Volume = (4/3)π * 8Next, I multiply 4/3 by 8. Let's see, 4 times 8 is 32, and then divided by 3, so that's 32/3. Therefore, the volume is:Volume = (32/3)πHmm, that seems right. Let me double-check my steps to make sure I didn't make any mistakes. Starting from the surface area:4πr² = 16πDivide both sides by π: 4r² = 16Divide both sides by 4: r² = 4Square root: r = 2Then volume: (4/3)π(2)³ = (4/3)π * 8 = 32/3 πYeah, that all checks out. So the volume of the sphere is 32/3 π.Just to make sure, I'll visualize it. If the radius is 2, then the sphere isn't too big. The surface area being 16π makes sense because 4πr² with r=2 gives 4π*4=16π. And the volume, being proportional to r³, should be larger, which it is—32/3 π is about 10.666π, which seems reasonable for a sphere with radius 2.I think I'm confident with this answer now.

💡Okay, so Roberto is trying to save money by raising chickens instead of buying eggs from the store. He has four different chickens, each with different costs and egg-laying rates. I need to figure out after how many weeks raising these chickens will be cheaper than buying eggs.First, let's list out all the information given:- **Chicken 1**: 25 initially, 4 eggs per week, 1.50 weekly feed- **Chicken 2**: 30 initially, 3 eggs per week, 1.30 weekly feed- **Chicken 3**: 22 initially, 5 eggs per week, 1.10 weekly feed- **Chicken 4**: 35 initially, 2 eggs per week, 0.90 weekly feedRoberto used to buy 1 dozen (12) eggs a week for 2.So, the first thing I need to do is calculate the total initial cost for all four chickens. That should be straightforward:25 + 30 + 22 + 35 = 112Okay, so he spent 112 to buy all four chickens.Next, I need to figure out the total weekly cost to feed all four chickens. Let's add up the weekly feed costs:1.50 + 1.30 + 1.10 + 0.90 = 4.80So, every week, Roberto spends 4.80 on feed.Now, let's look at the total number of eggs produced each week by all four chickens:4 (from Chicken 1) + 3 (from Chicken 2) + 5 (from Chicken 3) + 2 (from Chicken 4) = 14 eggs per weekHe used to buy 12 eggs a week for 2, but now he's getting 14 eggs. Since he's getting more eggs, it might be useful to consider the cost per egg to see if it's cheaper.But maybe a simpler approach is to compare the total cost of raising the chickens versus the total cost of buying eggs.Let's think about it in terms of total costs over time.The initial cost is 112, and then each week he spends 4.80 on feed. So, the total cost after W weeks is:Total Cost = 112 + (4.80 * W)On the other hand, if he continues buying eggs, he spends 2 per week. So, the total cost for buying eggs after W weeks is:Total Cost = 2 * WWe need to find the value of W where the total cost of raising chickens becomes less than the total cost of buying eggs.So, set up the inequality:112 + 4.80W < 2WWait, that doesn't seem right because 4.80 is more than 2, so the left side would always be greater. That can't be right because he's getting more eggs, so maybe I need to adjust the comparison.Actually, he's getting 14 eggs a week from the chickens, which is more than the 12 he used to buy. So, maybe we should consider the cost per dozen eggs.He used to spend 2 for 12 eggs. Now, he's getting 14 eggs. So, the cost per dozen from the chickens would be:Cost per dozen = (112 + 4.80W) / (14/12 * W)Wait, that seems complicated. Maybe another approach is better.Alternatively, since he's getting 14 eggs a week, and he used to buy 12, he's effectively saving the cost of 12 eggs, which was 2, and getting 2 extra eggs for free. So, his weekly savings are 2.But he has to cover the initial cost of 112 and the weekly feed cost of 4.80.So, the net weekly cost is 4.80 (feed) minus 2 (savings from not buying eggs) = 2.80 per week net cost.Now, to find out when the initial cost is covered by the net weekly cost:112 / 2.80 per week = 40 weeksSo, after 40 weeks, the chickens will have cost him the same as buying eggs, and after that, they will be cheaper.Wait, let me double-check that.Total cost after 40 weeks:112 + (4.80 * 40) = 112 + 192 = 304Total cost of buying eggs for 40 weeks:2 * 40 = 80Wait, that doesn't make sense because 304 is way more than 80. I must have made a mistake.Oh, I see. I think I confused the net cost. Let's try again.He saves 2 per week by not buying eggs, but he spends 4.80 per week on feed. So, his net weekly expenditure is 4.80 - 2 = 2.80 per week.Therefore, to cover the initial 112, it would take:112 / 2.80 = 40 weeksAfter 40 weeks, the total expenditure on chickens would be:112 + (4.80 * 40) = 112 + 192 = 304And the total cost of buying eggs for 40 weeks would be:2 * 40 = 80Wait, that still doesn't make sense because 304 is more than 80. I must be misunderstanding something.Perhaps I should consider that the 2 per week is the cost he avoids, so his net benefit is 2 per week, and he needs to cover the initial cost plus the feed cost.So, the equation should be:Initial Cost + (Feed Cost * W) = Savings * W112 + 4.80W = 2WBut solving for W:112 = 2W - 4.80W112 = -2.80WW = 112 / -2.80W = -40Negative weeks don't make sense, so I must have set up the equation incorrectly.Perhaps the correct equation is:Initial Cost + (Feed Cost * W) = Savings * W112 + 4.80W = 2WBut this leads to a negative W, which is impossible. So, maybe I need to rearrange it.Let's think about it differently. The total cost of raising chickens is 112 + 4.80W, and the total cost of buying eggs is 2W. We want to find when the chicken cost is less than the egg cost:112 + 4.80W < 2WBut 4.80W is greater than 2W, so the left side will always be greater. This suggests that raising chickens is never cheaper, which contradicts the initial thought.Wait, that can't be right because he's getting more eggs. Maybe I need to consider the value of the extra eggs.He gets 14 eggs instead of 12, so he's getting 2 extra eggs for free. The value of those extra eggs could be considered as additional savings.If he sells the extra 2 eggs, how much would that be? If 12 eggs cost 2, then 1 egg costs 2/12 = 0.1667.So, 2 extra eggs would be worth 2 * 0.1667 = 0.3334 per week.Therefore, his total savings per week would be 2 (from not buying 12 eggs) + 0.3334 (from selling extra 2 eggs) = 2.3334 per week.Now, his net weekly cost is 4.80 (feed) - 2.3334 (savings) = 2.4666 per week.To cover the initial 112:112 / 2.4666 ≈ 45.4 weeksSo, approximately 46 weeks.But this is getting complicated. Maybe the simplest way is to compare the total cost of chickens versus the total cost of eggs over W weeks.Total cost of chickens: 112 + 4.80WTotal cost of eggs: 2WWe want to find W where:112 + 4.80W < 2WBut as before, this leads to a negative W, which is impossible. Therefore, raising chickens is never cheaper in this scenario.Wait, that can't be right because he's getting more eggs. There must be a mistake in the approach.Perhaps the correct way is to calculate the cost per dozen from the chickens and compare it to the store price.He gets 14 eggs per week, which is 14/12 = 1.1667 dozens per week.The cost per dozen from chickens is:Total cost / dozens = (112 + 4.80W) / (1.1667W)We want this to be less than 2 per dozen.So:(112 + 4.80W) / (1.1667W) < 2Multiply both sides by 1.1667W:112 + 4.80W < 2 * 1.1667W112 + 4.80W < 2.3334WSubtract 4.80W from both sides:112 < 2.3334W - 4.80W112 < -2.4666WThis leads to a negative W again, which is impossible.This suggests that the cost per dozen from chickens will never be less than 2. Therefore, raising chickens is not cheaper than buying eggs in this scenario.But this contradicts the initial problem statement that asks after how many weeks the chickens will be cheaper. So, perhaps I'm missing something.Wait, maybe the initial cost is a one-time expense, and the weekly costs are ongoing. So, the total cost of chickens is 112 + 4.80W, and the total cost of eggs is 2W.We need to find when 112 + 4.80W < 2WBut as before, this leads to a negative W, which is impossible. Therefore, the chickens are never cheaper.But that can't be right because he's getting more eggs. Maybe the problem assumes that he only needs 12 eggs, so the extra 2 are not considered in the savings.In that case, his savings are still 2 per week, and his net weekly cost is 4.80 - 2 = 2.80 per week.To cover the initial 112:112 / 2.80 = 40 weeksSo, after 40 weeks, the chickens will have cost him the same as buying eggs, and after that, they will be cheaper.Therefore, the answer is 40 weeks.

💡First, I need to calculate the volume of one brick. The brick measures 20 cm in length, 10 cm in width, and 7.5 cm in height. Next, I'll determine the total volume of all the bricks by multiplying the volume of one brick by the total number of bricks, which is 27,000.To find the volume of the wall, I need to convert the wall's measurements from meters to centimeters. The wall is 27 meters long and 2 meters wide.I'll denote the height of the wall in centimeters as ( h_{text{cm}} ).Using the formula for the volume of the wall, I'll set the total volume of the bricks equal to the volume of the wall and solve for the height ( h_{text{cm}} ).Finally, I'll convert the height from centimeters to meters to get the height of the wall in the desired unit.

💡Okay, so I have this problem: I need to find out how many natural numbers from 1 to 2015 inclusive have a digit sum that's a multiple of 5. Hmm, okay, let me think about how to approach this.First, I remember that the digit sum of a number is just the sum of all its digits. For example, the digit sum of 123 is 1 + 2 + 3 = 6. So, I need to find all numbers where this sum is divisible by 5. That means the digit sum should be 0, 5, 10, 15, etc.Now, considering the range from 1 to 2015, that's a pretty big range. I wonder if there's a pattern or a systematic way to count these numbers without having to check each one individually. Maybe I can use some modular arithmetic or properties of numbers.I recall that for any set of consecutive numbers, the digit sums can cycle through different remainders when divided by 5. For example, if I take ten consecutive numbers, their digit sums modulo 5 will cover all possible remainders from 0 to 4, right? So, in any block of ten numbers, exactly two of them will have digit sums divisible by 5. That seems useful.Let me test this idea with a small example. Take numbers 1 to 10:1: 1 (sum=1)2: 2 (sum=2)3: 3 (sum=3)4: 4 (sum=4)5: 5 (sum=5) → divisible by 56: 6 (sum=6)7: 7 (sum=7)8: 8 (sum=8)9: 9 (sum=9)10: 1 + 0 = 1Wait, in this case, only 5 has a digit sum divisible by 5. Hmm, that's only one number. Maybe my initial thought was wrong. Let me check another set.Take numbers 10 to 19:10: 1 + 0 = 111: 1 + 1 = 212: 1 + 2 = 313: 1 + 3 = 414: 1 + 4 = 5 → divisible by 515: 1 + 5 = 616: 1 + 6 = 717: 1 + 7 = 818: 1 + 8 = 919: 1 + 9 = 10 → divisible by 5Okay, here we have two numbers: 14 and 19. So, in this block of ten, two numbers have digit sums divisible by 5. That seems to confirm my initial thought. Maybe the first block (1-10) is a special case because it includes single-digit numbers, which might affect the count.So, if I consider numbers from 10 onwards, every block of ten numbers will have exactly two numbers with digit sums divisible by 5. That makes sense because when you increment a number, the digit sum increases by 1 unless there's a carry-over, which can cause the digit sum to decrease. But in the case of numbers without trailing 9s, the digit sum just increases by 1 each time. So, in ten consecutive numbers, the digit sums modulo 5 will cycle through all residues, resulting in two numbers where the digit sum is 0 modulo 5.Given that, I can calculate how many such blocks there are between 1 and 2015 and then multiply by 2 to get the total count. But I also need to account for the numbers before 10 and the numbers beyond the last complete block up to 2015.Let me break it down step by step.1. **Numbers from 1 to 9:** These are single-digit numbers. Their digit sums are the numbers themselves. So, which of these are divisible by 5? Only 5. So, that's 1 number.2. **Numbers from 10 to 2015:** Let's find out how many numbers are there from 10 to 2015. That's 2015 - 10 + 1 = 2006 numbers. Now, how many blocks of ten are there in 2006 numbers? That's 2006 / 10 = 200.6. So, there are 200 complete blocks of ten and a partial block of 6 numbers. Each complete block of ten contributes 2 numbers with digit sums divisible by 5. So, 200 blocks * 2 = 400 numbers. Now, the partial block from 2010 to 2015 (since 2015 - 2010 + 1 = 6 numbers). Let's list these numbers and check their digit sums: - 2010: 2 + 0 + 1 + 0 = 3 - 2011: 2 + 0 + 1 + 1 = 4 - 2012: 2 + 0 + 1 + 2 = 5 → divisible by 5 - 2013: 2 + 0 + 1 + 3 = 6 - 2014: 2 + 0 + 1 + 4 = 7 - 2015: 2 + 0 + 1 + 5 = 8 So, only 2012 has a digit sum divisible by 5. That's 1 number.3. **Total count:** - Numbers from 1-9: 1 - Numbers from 10-2015: 400 (from complete blocks) + 1 (from partial block) = 401 Wait, but earlier I thought the partial block contributed 1 number, so total would be 1 + 401 = 402.But let me double-check the partial block. From 2010 to 2015, I have 6 numbers, and only 2012 has a digit sum divisible by 5. So, that's correct.However, I want to make sure that my initial assumption about the blocks is correct. Let me test another block to confirm.Take numbers 20 to 29:20: 2 + 0 = 221: 2 + 1 = 322: 2 + 2 = 423: 2 + 3 = 5 → divisible by 524: 2 + 4 = 625: 2 + 5 = 726: 2 + 6 = 827: 2 + 7 = 928: 2 + 8 = 10 → divisible by 529: 2 + 9 = 11Yes, 23 and 28 have digit sums divisible by 5. So, again, two numbers in a block of ten. That seems consistent.Another test: numbers 90 to 99:90: 9 + 0 = 991: 9 + 1 = 10 → divisible by 592: 9 + 2 = 1193: 9 + 3 = 1294: 9 + 4 = 1395: 9 + 5 = 1496: 9 + 6 = 15 → divisible by 597: 9 + 7 = 1698: 9 + 8 = 1799: 9 + 9 = 18Here, 91 and 96 have digit sums divisible by 5. Again, two numbers. So, my initial assumption holds.Therefore, my calculation seems correct. From 10 to 2015, there are 200 complete blocks of ten, each contributing 2 numbers, so 400 numbers. Then, the partial block from 2010 to 2015 contributes 1 number. Plus the single-digit number 5. So, total is 402.Wait, but let me think about the transition from 999 to 1000. Does that affect the digit sums? For example, 999 has a digit sum of 27, and 1000 has a digit sum of 1. So, the digit sum drops from 27 to 1, which is a change of -26. That's a significant drop, but in terms of modulo 5, 27 mod 5 is 2, and 1 mod 5 is 1. So, the digit sums modulo 5 go from 2 to 1, which is a decrease of 1. So, the pattern continues without any disruption in terms of modulo 5.Therefore, my earlier reasoning still holds. Each block of ten numbers, regardless of where they are in the number line, will have exactly two numbers with digit sums divisible by 5.So, I think my answer is correct: 402 numbers from 1 to 2015 have digit sums divisible by 5.

💡First, I'll define the total number of employees as ( E ), the number of men as ( M ), and the number of women as ( W ).Given that 55% of all employees are men, I can express the number of men as:[M = 0.55E]Since the remaining employees are women, the number of women is:[W = E - M = E - 0.55E = 0.45E]Next, 40% of the women attended the picnic, so the number of women who attended is:[0.40W = 0.40 times 0.45E = 0.18E]We know that 29% of all employees attended the picnic, which means:[0.29E text{ employees attended the picnic}]To find the number of men who attended the picnic, I'll subtract the number of women who attended from the total number of attendees:[text{Number of men who attended} = 0.29E - 0.18E = 0.11E]Finally, to determine the percentage of men who attended the picnic, I'll divide the number of men who attended by the total number of men and multiply by 100:[text{Percentage of men who attended} = left( frac{0.11E}{0.55E} right) times 100 = left( frac{0.11}{0.55} right) times 100 = 0.2 times 100 = 20%]

💡First, I need to understand the ratio of the hours worked by the three people, which is 1:2:3. This means that for every hour the first person worked, the second person worked twice as much, and the third person worked three times as much.Next, I'll assign a variable to represent the common multiplier in the ratio. Let's say the first person worked ( x ) hours. Then, the second person worked ( 2x ) hours, and the third person worked ( 3x ) hours.The total hours worked by all three people should add up to 120 hours. So, I can set up the equation:[x + 2x + 3x = 120]Combining like terms gives:[6x = 120]To find the value of ( x ), I'll divide both sides of the equation by 6:[x = frac{120}{6} = 20]Now that I know ( x = 20 ), I can determine the hours worked by each person:- The first person worked ( x = 20 ) hours.- The second person worked ( 2x = 40 ) hours.- The third person worked ( 3x = 60 ) hours.Finally, to find out how many more hours the hardest working person (third person) worked than the person who worked the least (first person), I'll subtract the hours worked by the first person from the hours worked by the third person:[60 - 20 = 40]