💡First, I need to determine how many hours Percy swims each day during the weekdays. He swims 1 hour before school and 1 hour after school, which totals 2 hours per weekday.Next, I'll calculate the total swimming hours for the weekdays over one week. Since there are 5 weekdays, multiplying the daily swimming hours by 5 gives 10 hours per week.Then, I'll add the swimming hours Percy does on the weekend. He swims 3 hours on the weekend, so adding this to the weekday total gives 13 hours of swimming per week.Finally, to find out how many hours Percy swims over 4 weeks, I'll multiply the weekly swimming total by 4. This results in 52 hours of swimming over 4 weeks.

💡Okay, so I have this equation to solve: 2 + 3iw = 4 - 2iw. Hmm, I need to find the value of w. Alright, let's break this down step by step. First, I remember that i is the imaginary unit where i squared equals -1. But in this equation, I don't see any i squared terms, so maybe I don't need to worry about that right now.Let me write down the equation again to make sure I have it right: 2 + 3iw = 4 - 2iw. So, I have constants and terms with w on both sides. I think the first thing I should do is get all the terms with w on one side and the constants on the other side. That way, I can solve for w more easily.Alright, so let's see. On the left side, I have 2 plus 3iw, and on the right side, I have 4 minus 2iw. Maybe I can add 2iw to both sides to get rid of the -2iw on the right. Let me try that. If I add 2iw to both sides, I get:2 + 3iw + 2iw = 4 - 2iw + 2iwSimplifying both sides, the right side becomes just 4 because -2iw and +2iw cancel each other out. On the left side, I have 2 plus (3iw + 2iw), which is 2 + 5iw. So now the equation is:2 + 5iw = 4Okay, that looks better. Now I need to get rid of the 2 on the left side. I can subtract 2 from both sides to do that. Let's do that:2 + 5iw - 2 = 4 - 2Simplifying both sides again, the left side becomes just 5iw, and the right side becomes 2. So now I have:5iw = 2Alright, now I need to solve for w. To do that, I can divide both sides by 5i. Let me write that down:w = 2 / (5i)Hmm, so w equals 2 divided by 5i. But I remember that having i in the denominator isn't usually preferred. I think I need to rationalize the denominator to make it look nicer. How do I do that? Oh, right, I can multiply both the numerator and the denominator by i to get rid of the i in the denominator.Let me try that. So, multiplying numerator and denominator by i:w = (2 * i) / (5i * i)Simplifying the denominator, i times i is i squared, which is -1. So the denominator becomes 5 * (-1) = -5. The numerator is 2i. So now I have:w = 2i / (-5)Which simplifies to:w = -2i / 5So, w equals negative 2i over 5. Let me just double-check my steps to make sure I didn't make any mistakes. Starting from the original equation, I moved the -2iw to the left side by adding 2iw, which gave me 5iw on the left. Then I subtracted 2 from both sides to isolate the term with w, resulting in 5iw = 2. Dividing both sides by 5i gave me w = 2/(5i), and then multiplying numerator and denominator by i to rationalize the denominator led me to w = -2i/5.Wait, let me check if I did the multiplication correctly. When I multiplied numerator and denominator by i, the numerator became 2i, and the denominator became 5i squared, which is 5*(-1) = -5. So yes, that seems correct. So, w equals -2i over 5.I think that's the solution. Let me just plug it back into the original equation to verify. If w = -2i/5, then let's compute both sides:Left side: 2 + 3i*(-2i/5) = 2 + (3i * -2i)/5 = 2 + (-6i²)/5. Since i² is -1, this becomes 2 + (-6*(-1))/5 = 2 + 6/5 = 2 + 1.2 = 3.2.Right side: 4 - 2i*(-2i/5) = 4 - (2i * -2i)/5 = 4 - (-4i²)/5. Again, i² is -1, so this becomes 4 - (-4*(-1))/5 = 4 - 4/5 = 4 - 0.8 = 3.2.Both sides equal 3.2, which is 16/5. So, yes, the solution w = -2i/5 satisfies the original equation. Therefore, I'm confident that my solution is correct.

💡Okay, so I have this problem here: The space diagonal of a cube has length 6. I need to find the surface area of the cube. Hmm, let me think about how to approach this.First, I remember that a cube has all sides equal, right? So, if I can find the length of one edge, I can figure out the surface area. But the problem gives me the space diagonal, not the edge length. I need to relate the space diagonal to the edge length.I recall that in a cube, the space diagonal can be found using the Pythagorean theorem in three dimensions. If the edge length is 's', then the space diagonal 'd' is given by the formula:d = s√3Wait, is that right? Let me visualize the cube. The space diagonal goes from one corner of the cube to the opposite corner, passing through the interior of the cube. So, if I consider the diagonal of one face, that would be s√2, and then the space diagonal would involve that face diagonal and the edge perpendicular to it. So, using the Pythagorean theorem again, the space diagonal would be sqrt((s√2)^2 + s^2) = sqrt(2s^2 + s^2) = sqrt(3s^2) = s√3. Yeah, that seems correct.So, given that the space diagonal is 6, I can set up the equation:6 = s√3To find 's', I can divide both sides by √3:s = 6 / √3Hmm, but having a square root in the denominator isn't usually preferred. I think I need to rationalize the denominator. To do that, I can multiply both the numerator and the denominator by √3:s = (6 * √3) / (√3 * √3) = (6√3) / 3 = 2√3Okay, so the edge length 's' is 2√3. Now, I need to find the surface area of the cube. The surface area of a cube is calculated by the formula:Surface Area = 6s²Because a cube has 6 faces, each with an area of s².So, substituting the value of 's' we found:Surface Area = 6 * (2√3)²Let me compute (2√3)² first. That would be:(2√3)² = 2² * (√3)² = 4 * 3 = 12So, Surface Area = 6 * 12 = 72Wait, that seems straightforward. Let me double-check my steps to make sure I didn't make a mistake.1. I started with the space diagonal formula: d = s√3.2. Plugged in d = 6: 6 = s√3.3. Solved for 's': s = 6 / √3, then rationalized to get s = 2√3.4. Calculated the surface area using 6s²: 6*(2√3)² = 6*12 = 72.Everything seems to check out. I don't think I made any calculation errors. So, the surface area of the cube should be 72.**Final Answer**The surface area of the cube is boxed{72}.

💡Okay, so Nathan is planning his wedding reception and needs to figure out how much it's going to cost for all the decorations. He has 30 tables to set up, and each table needs a few things: a linen tablecloth, 6 place settings, and a centerpiece. Plus, there's some lighting for the hall that costs 450. First, I need to break down each component and calculate the costs, making sure to apply any discounts or additional charges where necessary. Let's start with the tablecloths. Each tablecloth costs 25 to rent, but there's a 15% service charge on top of that. So, I need to calculate the service charge and add it to the rental cost.Next, the place settings. Each table needs 6 place settings, and each place setting costs 12. That means for one table, it's 6 times 12, and then I need to multiply that by 30 tables to get the total cost for all place settings.Now, the centerpieces are a bit more complicated because they involve different types of flowers, each with their own costs and discounts. Each centerpiece has 15 roses, 20 lilies, 5 daisies, and 3 sunflowers. Roses cost 6 each with a 10% discount for bulk purchase, lilies are 5 each with a 5% discount, daisies are 3 each with no discount, and sunflowers are 4 each with no discount. I need to calculate the cost for each type of flower, apply the discounts where applicable, and then sum them up for one centerpiece. After that, I'll multiply by 30 to get the total cost for all centerpieces.Finally, there's the lighting cost, which is a flat 450. Once I have all these individual costs calculated, I'll add them all together to get the total decoration cost. I need to make sure I don't miss any discounts or additional charges and that I apply them correctly to avoid overestimating or underestimating the total cost.Let me start by calculating the cost for the tablecloths. Each tablecloth is 25, and there's a 15% service charge. So, 15% of 25 is 3.75. Adding that to the 25 gives me 28.75 per tablecloth. Since there are 30 tables, I'll multiply 28.75 by 30 to get the total cost for all tablecloths.Next, the place settings. Each place setting is 12, and each table needs 6 of them. So, 6 times 12 is 72 per table. For 30 tables, that's 72 multiplied by 30.Now, onto the centerpieces. Starting with roses: 15 roses at 6 each would be 90, but there's a 10% discount. 10% of 90 is 9, so the discounted cost for roses is 81. For lilies: 20 lilies at 5 each is 100, with a 5% discount. 5% of 100 is 5, so the discounted cost for lilies is 95.Daisies are 5 at 3 each, which is 15, and sunflowers are 3 at 4 each, which is 12. Adding all these together: 81 for roses, 95 for lilies, 15 for daisies, and 12 for sunflowers gives me a total of 203 per centerpiece. Multiplying that by 30 tables gives the total cost for all centerpieces.Finally, adding the lighting cost of 450.Now, I'll sum up all these costs: tablecloths, place settings, centerpieces, and lighting. This should give me the total decoration cost after applying all discounts and additional charges.I need to make sure I do each calculation step by step to avoid any mistakes. Let me double-check each part to ensure accuracy.

💡Okay, so I need to figure out if the function ( g(x) = 3^{x^2 - 3} - |x| + sin(x) ) is even, odd, or neither. Hmm, I remember that even functions satisfy ( g(-x) = g(x) ) for all ( x ) in their domain, and odd functions satisfy ( g(-x) = -g(x) ). If neither of these conditions hold, then the function is neither even nor odd.Alright, let me start by finding ( g(-x) ). That means I'll substitute ( -x ) into the function wherever there's an ( x ). So, let's do that step by step.First, the term ( 3^{x^2 - 3} ). If I replace ( x ) with ( -x ), it becomes ( 3^{(-x)^2 - 3} ). Wait, ( (-x)^2 ) is the same as ( x^2 ), right? So, that term simplifies to ( 3^{x^2 - 3} ). Okay, so the first term is the same as in ( g(x) ).Next, the term ( -|x| ). If I replace ( x ) with ( -x ), it becomes ( -|-x| ). But the absolute value of ( -x ) is the same as the absolute value of ( x ), so this term becomes ( -|x| ). So, the second term is also the same as in ( g(x) ).Now, the last term is ( sin(x) ). If I replace ( x ) with ( -x ), it becomes ( sin(-x) ). I remember that sine is an odd function, so ( sin(-x) = -sin(x) ). Therefore, this term becomes ( -sin(x) ).Putting it all together, ( g(-x) = 3^{x^2 - 3} - |x| - sin(x) ).Now, let's compare ( g(-x) ) with ( g(x) ). The original function is ( g(x) = 3^{x^2 - 3} - |x| + sin(x) ). So, ( g(-x) ) is ( 3^{x^2 - 3} - |x| - sin(x) ).Comparing term by term:- The first term ( 3^{x^2 - 3} ) is the same in both.- The second term ( -|x| ) is the same in both.- The third term in ( g(x) ) is ( +sin(x) ), while in ( g(-x) ) it's ( -sin(x) ).So, ( g(-x) ) is not equal to ( g(x) ) because the sine term has the opposite sign. That means it's not an even function.Now, let's check if it's odd. An odd function would satisfy ( g(-x) = -g(x) ). Let's compute ( -g(x) ):- ( -g(x) = -3^{x^2 - 3} + |x| - sin(x) ).Comparing this with ( g(-x) = 3^{x^2 - 3} - |x| - sin(x) ), they are not the same. The first terms have opposite signs, the second terms have opposite signs, and the third terms are the same. So, ( g(-x) ) is not equal to ( -g(x) ) either.Therefore, since ( g(-x) ) is neither equal to ( g(x) ) nor ( -g(x) ), the function ( g(x) ) is neither even nor odd.Wait, let me double-check to make sure I didn't make a mistake. I substituted ( -x ) into each term correctly, right? The exponent ( x^2 ) remains the same because squaring a negative gives a positive. The absolute value also remains the same. The sine term flips sign because sine is odd. So, yes, ( g(-x) ) ends up being ( 3^{x^2 - 3} - |x| - sin(x) ), which is different from both ( g(x) ) and ( -g(x) ).Yeah, I think that's correct. So, the function is neither even nor odd.

💡First, I need to determine the investment periods and amounts for both A and B throughout the year.A initially invested Rs. 3000 for the first 8 months and then withdrew Rs. 1000, leaving an investment of Rs. 2000 for the remaining 4 months.B initially invested Rs. 4000 for the first 8 months and then added Rs. 1000, increasing the investment to Rs. 5000 for the remaining 4 months.Next, I calculate the total investment-months for each partner to understand their contribution over the year.For A:- 3000 Rs. * 8 months = 24000 investment-months- 2000 Rs. * 4 months = 8000 investment-months- Total = 32000 investment-monthsFor B:- 4000 Rs. * 8 months = 32000 investment-months- 5000 Rs. * 4 months = 20000 investment-months- Total = 52000 investment-monthsNow, I establish the ratio of their investments:A : B = 32000 : 52000 = 8 : 13Given that A's share of the profit is Rs. 320, I can use the ratio to find the total profit.Let the total profit be P.A's share of the profit = (8/21) * P320 = (8/21) * PP = 320 * (21/8)P = 40 * 21P = Rs. 840Therefore, the total profit at the end of the year is Rs. 840.