💡First, I need to determine the total weight contributed by the envelopes alone. Each envelope weighs 8.5 grams, and there are 880 envelopes. So, I'll multiply the weight of one envelope by the number of envelopes to find the total weight of all envelopes.Next, I'll calculate the total weight of the additional postal materials, which include stamps, labels, and seals. Each envelope has an additional 2 grams of these materials. Therefore, I'll multiply the additional weight per envelope by the total number of envelopes to find the total weight of all additional materials.Finally, I'll add the total weight of the envelopes and the total weight of the additional materials together to find the overall total weight of the mailing.

💡First, identify the total number of marbles Susan has, which is 15. Among these, there are 5 specific colored marbles (red, green, blue, yellow, and purple) and one black marble.Susan needs to choose 5 marbles with the following conditions:1. Exactly one of the chosen marbles must be one of the specific colors (red, green, blue, yellow, or purple).2. None of the chosen marbles can be black.To satisfy these conditions, we can break down the problem into two steps:1. **Choosing one specific colored marble:** There are 5 specific colored marbles to choose from, so there are 5 ways to select one of them.2. **Choosing the remaining 4 marbles:** After selecting one specific colored marble, there are 15 - 1 (black) - 5 (specific colors) = 9 marbles left. Susan needs to choose 4 marbles from these 9, which can be calculated using the combination formula ( binom{9}{4} ).Calculating ( binom{9}{4} ):[binom{9}{4} = frac{9!}{4!(9-4)!} = frac{9 times 8 times 7 times 6}{4 times 3 times 2 times 1} = 126]Finally, multiply the number of ways to choose the specific colored marble by the number of ways to choose the remaining marbles:[5 times 126 = 630]

💡First, I need to understand the problem: there's a wooden stick that's 11 meters long, and it's broken into two pieces. I need to find the probability that both pieces are at least 3 meters long.To approach this, I'll consider the possible points where the stick can be broken. Since the stick is 11 meters long, the break can happen anywhere along its length.Next, I'll determine the range of valid break points that ensure both pieces are no less than 3 meters. If I break the stick at a point that's at least 3 meters from one end, the other piece will automatically be at least 3 meters as well because the total length is 11 meters.So, the valid break points are from 3 meters to 8 meters along the stick. This gives me a valid range of 5 meters.Finally, to find the probability, I'll divide the length of the valid range by the total length of the stick. This gives me the probability that both pieces will be no less than 3 meters in length.

💡To determine the number of possible integer side lengths for the third side of a triangle with sides measuring 8 units and 3 units, I start by applying the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.First, I consider the sum of the two known sides: 8 + 3 = 11. According to the theorem, the third side must be less than this sum, so the third side must be less than 11 units.Next, I examine the difference between the two known sides: 8 - 3 = 5. The third side must be greater than this difference to satisfy the Triangle Inequality, meaning the third side must be greater than 5 units.Combining these two conditions, the third side must be greater than 5 units and less than 11 units. Since we are looking for integer side lengths, the possible values for the third side are 6, 7, 8, 9, and 10 units.Counting these integer values, there are 5 possible integer side lengths that satisfy the conditions of the Triangle Inequality Theorem.

💡Okay, so I've got this problem about an arithmetic sequence. Let me try to understand what it's asking. It says that in the arithmetic sequence {a_n}, the sixth term, a_6, is equal to 2. I need to find the sum of the first 11 terms, which is denoted by S_11. The options given are A: 44, B: 33, C: 22, and D: 11.Alright, let's start by recalling what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, usually denoted by 'd'. So, if I know the first term, a_1, and the common difference, d, I can find any term in the sequence using the formula:a_n = a_1 + (n - 1)dGiven that a_6 = 2, I can plug that into the formula:a_6 = a_1 + (6 - 1)d2 = a_1 + 5dSo, I have this equation: a_1 + 5d = 2. That's one equation, but I don't know the values of a_1 or d. Hmm, so I need another equation to solve for both variables. But wait, the problem doesn't give me any more terms. It just asks for the sum of the first 11 terms. Maybe I don't need to find a_1 and d individually?Let me think about the formula for the sum of the first n terms of an arithmetic sequence. The sum S_n is given by:S_n = n/2 * (a_1 + a_n)Alternatively, it can also be written as:S_n = n/2 * [2a_1 + (n - 1)d]Since I need to find S_11, let's write that out:S_11 = 11/2 * (a_1 + a_11)But I don't know a_11 either. Wait, can I express a_11 in terms of a_1 and d? Yes, using the nth term formula:a_11 = a_1 + (11 - 1)da_11 = a_1 + 10dSo, substituting back into the sum formula:S_11 = 11/2 * (a_1 + a_1 + 10d)S_11 = 11/2 * (2a_1 + 10d)S_11 = 11/2 * 2(a_1 + 5d)S_11 = 11 * (a_1 + 5d)But from earlier, I know that a_1 + 5d = 2. So, substituting that in:S_11 = 11 * 2S_11 = 22So, the sum of the first 11 terms is 22. Let me check the options: C is 22, so that must be the answer.Wait, let me make sure I didn't skip any steps or make any mistakes. I used the formula for the nth term to express a_6 in terms of a_1 and d, then used the sum formula for S_11, expressed a_11 in terms of a_1 and d, substituted back, and then noticed that a_1 + 5d is exactly a_6, which is given as 2. So, multiplying 11 by 2 gives 22. That seems correct.I don't think I made any calculation errors. The key insight was recognizing that in the sum formula, the term a_1 + a_11 simplifies to 2a_6 because a_11 is a_6 plus 5d, and a_1 is a_6 minus 5d. So, when you add them together, the d terms cancel out, leaving 2a_6. That's a useful property of arithmetic sequences.Alternatively, I could have thought about the average of the first and the 11th term. Since the average of the first and last term in an arithmetic sequence is equal to the average of all the terms, and there are 11 terms, the sum would be 11 times that average. The average of a_1 and a_11 is (a_1 + a_11)/2, which is equal to a_6 because in an arithmetic sequence, the middle term is the average. So, the average is 2, and multiplying by 11 gives 22. That's another way to see it.Yeah, I think that's solid. I don't see any issues with the reasoning. So, I'm confident that the answer is 22.

💡Okay, so I have this problem here about a rectangular solid with length, width, and height of 1, 2, and 3 respectively, and it's circumscribed by a sphere. I need to find the volume of that sphere. Hmm, okay, let me think about how to approach this.First, I remember that when a rectangular solid is circumscribed by a sphere, the sphere touches all the corners of the solid. That means the sphere's diameter is equal to the space diagonal of the rectangular solid. So, if I can find the length of the space diagonal, I can find the diameter of the sphere, and then the radius, which is half of that. Once I have the radius, I can use the formula for the volume of a sphere, which is (4/3)πr³.Alright, let's break it down step by step.1. **Find the space diagonal of the rectangular solid.** I recall that the formula for the space diagonal (d) of a rectangular solid with length (l), width (w), and height (h) is: [ d = sqrt{l^2 + w^2 + h^2} ] Plugging in the given dimensions: [ d = sqrt{1^2 + 2^2 + 3^2} = sqrt{1 + 4 + 9} = sqrt{14} ] So, the space diagonal is √14. That means the diameter of the sphere is √14.2. **Find the radius of the sphere.** Since the diameter is √14, the radius (r) is half of that: [ r = frac{sqrt{14}}{2} ] 3. **Calculate the volume of the sphere.** The formula for the volume (V) of a sphere is: [ V = frac{4}{3} pi r^3 ] Substituting the radius we found: [ V = frac{4}{3} pi left( frac{sqrt{14}}{2} right)^3 ] Let me compute that step by step. First, cube the radius: [ left( frac{sqrt{14}}{2} right)^3 = frac{(sqrt{14})^3}{2^3} = frac{14 sqrt{14}}{8} ] Simplify that: [ frac{14 sqrt{14}}{8} = frac{7 sqrt{14}}{4} ] Now, multiply by (4/3)π: [ V = frac{4}{3} pi times frac{7 sqrt{14}}{4} ] The 4 in the numerator and denominator cancels out: [ V = frac{7 sqrt{14}}{3} pi ] So, the volume of the sphere is (7√14)/3 π.4. **Check the answer options.** Looking back at the options provided: A: 14π B: 28π C: (7√14)/3 π D: (14√14)/3 π The result I got matches option C.Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting from the space diagonal:- 1² + 2² + 3² = 1 + 4 + 9 = 14. So, √14 is correct for the space diagonal.Radius is half of that, so √14 / 2. That seems right.Volume formula: (4/3)πr³. Plugging in √14 / 2:First, (√14 / 2)³ = (14^(1/2))³ / 8 = 14^(3/2) / 8 = (14 * √14) / 8.Simplify: (14 / 8) * √14 = (7/4) * √14.Then, (4/3)π * (7/4)√14. The 4 cancels out, leaving (7/3)√14 π.Yes, that's correct. So, option C is indeed the right answer.I think I've got it. The volume of the sphere is (7√14)/3 π.**Final Answer**boxed{C}