💡Alright, so I have this problem about an infinite geometric series. It says the first term is 540, and the sum of the series is 4500. I need to find the common ratio and the second term of the series. Hmm, okay, let's break this down step by step.First, I remember that for an infinite geometric series, the sum can be found using the formula:[ S = frac{a}{1 - r} ]where ( S ) is the sum, ( a ) is the first term, and ( r ) is the common ratio. I also recall that this formula only works if the absolute value of ( r ) is less than 1, so ( |r| < 1 ). That makes sense because if the common ratio is too big, the series won't converge to a finite sum.Okay, so in this problem, ( a = 540 ) and ( S = 4500 ). I need to find ( r ). Let's plug these values into the formula:[ 4500 = frac{540}{1 - r} ]Hmm, now I need to solve for ( r ). Let's rearrange the equation. First, multiply both sides by ( 1 - r ) to get rid of the denominator:[ 4500(1 - r) = 540 ]Now, let's divide both sides by 4500 to isolate ( 1 - r ):[ 1 - r = frac{540}{4500} ]Let me compute that fraction. 540 divided by 4500. Hmm, 540 divided by 4500 is the same as 54 divided by 450, which simplifies to 6 divided by 50, and that's 0.12. So,[ 1 - r = 0.12 ]Now, to find ( r ), I'll subtract 0.12 from 1:[ r = 1 - 0.12 = 0.88 ]Okay, so the common ratio ( r ) is 0.88. That seems reasonable since it's less than 1, so the series does converge.Now, the second part of the problem asks for the second term of the series. In a geometric series, each term is the previous term multiplied by the common ratio. So, the second term ( a_2 ) is:[ a_2 = a times r ]Plugging in the values we have:[ a_2 = 540 times 0.88 ]Let me calculate that. 540 times 0.88. Hmm, 540 times 0.8 is 432, and 540 times 0.08 is 43.2. Adding those together, 432 + 43.2 equals 475.2. So,[ a_2 = 475.2 ]Just to make sure I didn't make any mistakes, let me double-check my calculations. Starting with the sum formula:[ S = frac{a}{1 - r} ][ 4500 = frac{540}{1 - r} ]Solving for ( r ):[ 1 - r = frac{540}{4500} = 0.12 ][ r = 1 - 0.12 = 0.88 ]That looks correct. Then, calculating the second term:[ 540 times 0.88 = 475.2 ]Yep, that seems right too. So, I think I've got the correct values for both the common ratio and the second term.**Final Answer**The common ratio is boxed{0.88} and the second term is boxed{475.2}.

💡Okay, so I have this problem from "The Mathematical Classic of Sunzi," which is an ancient Chinese text. It's about a piece of wood and a rope. The problem goes like this: "There is a piece of wood whose length is unknown. When measured with a rope, the rope exceeds by 4.5 feet; when the rope is folded in half and used to measure the wood, it falls short by 1 foot. What is the length of the wood?" Alright, let's break this down. They want me to find the length of the wood, and they've given me two different scenarios involving a rope. I need to set up a system of equations based on these scenarios. They've also given me four options to choose from, labeled A to D. Each option presents a different system of equations. My job is to figure out which system correctly represents the problem.First, let's assign variables to make things clearer. Let's say the length of the rope is ( x ) feet, and the length of the wood is ( y ) feet. That seems straightforward enough.Now, let's look at the first condition: "When measured with a rope, the rope exceeds by 4.5 feet." So, if I use the rope to measure the wood, the rope is longer than the wood by 4.5 feet. That means the rope is 4.5 feet longer than the wood. In equation form, that would be:( x = y + 4.5 )Alternatively, rearranged, it's:( x - y = 4.5 )Okay, that's the first equation. Now, the second condition: "When the rope is folded in half and used to measure the wood, it falls short by 1 foot." So, if I fold the rope in half, its length becomes ( frac{x}{2} ). When I use this folded rope to measure the wood, it's not long enough by 1 foot. That means the folded rope is 1 foot shorter than the wood. So, in equation form, that would be:( frac{x}{2} = y - 1 )Or, rearranged:( frac{x}{2} + 1 = y )So, putting these two equations together, my system of equations should be:1. ( x - y = 4.5 )2. ( frac{x}{2} + 1 = y )Now, let's look at the options provided:A: ( left{begin{array}{l}{x - y = 4.5}{frac{1}{2}x + 1 = y}end{array}right. )B: ( left{begin{array}{l}{x - y = 4.5}{2x + 1 = y}end{array}right. )C: ( left{begin{array}{l}{x - y = 4.5}{frac{1}{2}x - 1 = y}end{array}right. )D: ( left{begin{array}{l}{y - x = 4.5}{2x - 1 = y}end{array}right. )Comparing my derived system with the options, option A matches exactly. The first equation is ( x - y = 4.5 ), and the second equation is ( frac{1}{2}x + 1 = y ). Just to be thorough, let's check why the other options are incorrect.Option B has the second equation as ( 2x + 1 = y ). That would imply that folding the rope in half makes it longer, which doesn't make sense because folding it in half should make it shorter. So, this is incorrect.Option C has the second equation as ( frac{1}{2}x - 1 = y ). This suggests that when folded, the rope is 1 foot shorter than half the rope's length, which doesn't align with the problem statement. The problem says the folded rope falls short by 1 foot, meaning the wood is longer by 1 foot, so it should be ( frac{1}{2}x + 1 = y ). Therefore, option C is also incorrect.Option D has the first equation as ( y - x = 4.5 ), which implies the wood is longer than the rope by 4.5 feet. But according to the problem, the rope exceeds the wood by 4.5 feet, so this is the opposite and incorrect. The second equation in D is ( 2x - 1 = y ), which again suggests the rope is longer when folded, which is not the case. So, option D is incorrect as well.Therefore, after analyzing all options, option A is the correct system of equations that represents the problem accurately.

💡First, calculate the total area of the plantation by multiplying the length and width.Next, determine the total grams of peanuts produced by multiplying the area by the yield per square foot.Then, find out how much peanut butter can be made by using the conversion rate from peanuts to peanut butter.Convert the total grams of peanut butter into kilograms.Finally, calculate the total revenue by multiplying the total kilograms of peanut butter by the selling price per kilogram.

💡Okay, so I have this problem about a convex polyhedron Q. It has 30 vertices, 72 edges, and 44 faces. Out of these faces, 30 are triangular and 14 are quadrilateral. I need to find the number of space diagonals in Q. Hmm, space diagonals... I remember that in polyhedrons, a space diagonal is a line segment connecting two non-adjacent vertices that isn't an edge or a face diagonal. So, I need to figure out how many such diagonals exist.First, let me recall the formula for the number of diagonals in a polyhedron. I think it's similar to the number of diagonals in a polygon but extended to three dimensions. In a polygon with n sides, the number of diagonals is n(n-3)/2. But in a polyhedron, it's a bit different because we have to consider both edges and face diagonals.Wait, actually, for a polyhedron, the total number of line segments connecting any two vertices is given by the combination formula C(n, 2), where n is the number of vertices. So, for Q, which has 30 vertices, the total number of line segments would be C(30, 2). Let me calculate that:C(30, 2) = 30*29/2 = 435.So, there are 435 possible line segments connecting any two vertices. Now, out of these, 72 are edges of the polyhedron. So, if I subtract the edges, I get the number of diagonals, both face diagonals and space diagonals. Let me do that:435 - 72 = 363.So, there are 363 diagonals in total. But this includes both face diagonals and space diagonals. I need to find only the space diagonals, so I have to subtract the face diagonals from this number.Now, how do I find the number of face diagonals? Well, each face is either a triangle or a quadrilateral. Triangular faces don't have any diagonals because all their vertices are connected by edges. So, each triangular face contributes 0 diagonals. On the other hand, each quadrilateral face has two diagonals. So, for each quadrilateral face, there are two face diagonals.Given that there are 14 quadrilateral faces, the total number of face diagonals would be 14*2 = 28.So, subtracting these face diagonals from the total diagonals, I get:363 - 28 = 335.Therefore, the number of space diagonals in the polyhedron Q is 335.Wait, let me double-check my steps to make sure I didn't make a mistake. First, I calculated the total number of line segments between vertices, which is correct. Then, I subtracted the edges, which gives me all diagonals, both face and space. Then, I calculated the face diagonals by considering only the quadrilateral faces since triangles don't contribute any. That seems right because each quadrilateral has two diagonals. So, 14 quadrilaterals give 28 face diagonals. Subtracting that from 363 gives me 335 space diagonals. Yeah, that seems correct.I wonder if there's another way to approach this problem, maybe using Euler's formula or something related to the structure of the polyhedron. Euler's formula states that for any convex polyhedron, the number of vertices minus the number of edges plus the number of faces equals 2. Let me check if that holds here:30 - 72 + 44 = 2. Yes, 30 - 72 is -42, plus 44 is 2. So Euler's formula holds, which is a good consistency check.But does that help me find the number of space diagonals? Maybe not directly, but it's good to know that the given numbers are consistent with a convex polyhedron.Another thought: sometimes, in polyhedrons, especially those with all triangular faces, the number of space diagonals can be calculated using a different formula, but in this case, since there are both triangular and quadrilateral faces, I think my initial approach is still valid.Just to recap:1. Total line segments: C(30, 2) = 435.2. Subtract edges: 435 - 72 = 363 (all diagonals, face and space).3. Subtract face diagonals: 14 quadrilaterals * 2 diagonals each = 28.4. Space diagonals: 363 - 28 = 335.Yes, that seems solid. I don't see any errors in this reasoning. So, I'm confident that the number of space diagonals in Q is 335.

💡First, I need to determine the probability of drawing a white marble from Bag A and then a yellow marble from Bag B. Bag A has 3 white marbles out of a total of 7 marbles, so the probability of drawing a white marble is 3/7. Bag B contains 6 yellow marbles out of 10 marbles, so the probability of drawing a yellow marble from Bag B is 6/10. Multiplying these probabilities gives (3/7) * (6/10) = 18/70.Next, I calculate the probability of drawing a black marble from Bag A and then a yellow marble from Bag C. Bag A has 4 black marbles out of 7, so the probability of drawing a black marble is 4/7. Bag C has 2 yellow marbles out of 7 marbles, so the probability of drawing a yellow marble from Bag C is 2/7. Multiplying these probabilities gives (4/7) * (2/7) = 8/49.To find the total probability of drawing a yellow marble as the second marble, I add the probabilities of the two scenarios: 18/70 + 8/49. To add these fractions, I need a common denominator, which is 245. Converting both fractions, I get 18/70 = 63/245 and 8/49 = 40/245. Adding these together, I get 63/245 + 40/245 = 103/245.Therefore, the probability that the second marble drawn is yellow is 103/245.

💡First, I need to determine the dimensions of the mirror. The frame has a width of 15 cm on each side, so I will subtract twice this width from both the length and the width of the outer perimeter to find the mirror's dimensions.For the length:70 cm (outer length) minus 2 times 15 cm (frame width) equals 40 cm.For the width:90 cm (outer width) minus 2 times 15 cm (frame width) equals 60 cm.Now that I have the mirror's dimensions, I can calculate its area by multiplying the length by the width:40 cm multiplied by 60 cm equals 2400 cm².