answer：To find the projection vector of (overrightarrow{a}) onto (overrightarrow{b}), I need to follow these steps:1. **Calculate the dot product of (overrightarrow{a}) and (overrightarrow{b}):** [ overrightarrow{a} cdot overrightarrow{b} = (2)(2) + (-1)(2) + (2)(1) = 4 - 2 + 2 = 4 ]2. **Find the magnitude of (overrightarrow{b}):** [ |overrightarrow{b}| = sqrt{2^2 + 2^2 + 1^2} = sqrt{4 + 4 + 1} = sqrt{9} = 3 ]3. **Compute the projection vector using the formula:** [ text{proj}_{overrightarrow{b}} overrightarrow{a} = left( frac{overrightarrow{a} cdot overrightarrow{b}}{|overrightarrow{b}|^2} right) overrightarrow{b} = left( frac{4}{9} right)(2, 2, 1) = left( frac{8}{9}, frac{8}{9}, frac{4}{9} right) ]By following these steps, I arrive at the projection vector.

answer：Okay, so I have this trigonometry problem here, and I need to figure out the value of cos β. Let me read it again to make sure I understand all the details.It says that α and β are acute angles. That means both α and β are between 0 and 90 degrees, or in radians, between 0 and π/2. Good, so all their trigonometric functions will be positive because in the first quadrant, all functions are positive.We are given that cos α is equal to √5 / 5. Hmm, okay. So cos α = √5 / 5. I can maybe find sin α from this because I know that sin²α + cos²α = 1. Let me calculate that.So, sin²α = 1 - cos²α = 1 - (√5 / 5)² = 1 - (5 / 25) = 1 - 1/5 = 4/5. Therefore, sin α is √(4/5) which is 2√5 / 5. Since α is acute, sin α is positive, so that's straightforward.Next, we are told that sin(α + β) = 3/5. So, the sine of the sum of α and β is 3/5. Since both α and β are acute, their sum α + β can be at most π (180 degrees) because each is less than π/2. But since 3/5 is positive and less than 1, α + β must be in the first or second quadrant. However, since both α and β are acute, their sum is between 0 and π. But 3/5 is greater than 1/2, so sin(α + β) = 3/5 implies that α + β is in the first or second quadrant. But wait, since α is already greater than π/6 because cos α = √5 / 5 is less than 1/2, which is approximately 0.447, so cos α is about 0.447, which is less than 0.5, so α is greater than 60 degrees or π/3 radians.Wait, hold on. Let me think again. If cos α = √5 / 5, which is approximately 0.447, which is less than 0.5, so that would mean α is greater than 60 degrees because cosine decreases as the angle increases in the first quadrant. So, α is between 60 degrees and 90 degrees.Similarly, sin(α + β) = 3/5 is approximately 0.6, which is greater than 1/2, so α + β is between 36.87 degrees and 143.13 degrees. But since α is already greater than 60 degrees, α + β must be greater than 60 + 0 = 60 degrees, but since β is also acute, the maximum α + β can be is 90 + 90 = 180, but since both are acute, it's less than 180. But given that sin(α + β) is positive, it can be in the first or second quadrant.But wait, if α is greater than 60 degrees, and β is acute, then α + β could be greater than 90 degrees. So, α + β could be in the second quadrant, meaning that cos(α + β) would be negative. That's an important point because when we use the sine and cosine of α + β, we need to know the sign.So, let me write down what I know:1. cos α = √5 / 5, so sin α = 2√5 / 5.2. sin(α + β) = 3/5, so cos(α + β) can be found using the Pythagorean identity. Since sin²(α + β) + cos²(α + β) = 1, so cos²(α + β) = 1 - (9/25) = 16/25, so cos(α + β) is either 4/5 or -4/5.But since α + β could be in the first or second quadrant. If α + β is in the first quadrant, cos(α + β) is positive, and if it's in the second quadrant, it's negative.But from earlier, since α is greater than 60 degrees, and β is acute, so α + β could be greater than 90 degrees. Let's check the possible values.If α is greater than 60 degrees, and β is acute, say 30 degrees, then α + β could be 90 degrees or more. So, it's possible that α + β is in the second quadrant, so cos(α + β) is negative.Therefore, cos(α + β) = -4/5.Wait, but is that necessarily the case? Let me think. If α is greater than 60 degrees, and β is acute, so the smallest α + β can be is α + 0, which is α, which is greater than 60 degrees, so α + β is greater than 60 degrees. The maximum is α + β < 180 degrees, but since both are acute, α + β < 180. So, α + β is between 60 and 180 degrees.But sin(α + β) = 3/5. So, sin(α + β) = 3/5 is approximately 0.6, which corresponds to angles around 36.87 degrees and 143.13 degrees. But since α + β is greater than 60 degrees, the only possible angle is 143.13 degrees, which is in the second quadrant. Therefore, cos(α + β) must be negative, so cos(α + β) = -4/5.Okay, so now I have:cos(α + β) = -4/5sin(α + β) = 3/5sin α = 2√5 / 5cos α = √5 / 5Now, I need to find cos β.Hmm, how can I express cos β in terms of these known quantities?Well, β can be written as (α + β) - α. So, β = (α + β) - α.Therefore, cos β = cos[(α + β) - α] = cos(α + β)cos α + sin(α + β)sin α.That's the cosine of a difference formula. So, cos(A - B) = cos A cos B + sin A sin B.So, substituting the values we have:cos β = cos(α + β)cos α + sin(α + β)sin αPlugging in the known values:cos β = (-4/5)(√5 / 5) + (3/5)(2√5 / 5)Let me compute each term separately.First term: (-4/5)(√5 / 5) = (-4√5) / 25Second term: (3/5)(2√5 / 5) = (6√5) / 25Now, adding these two terms together:cos β = (-4√5 / 25) + (6√5 / 25) = (2√5) / 25So, cos β = 2√5 / 25.Wait, but let me double-check my calculations to make sure I didn't make any mistakes.First, cos(α + β) is -4/5, correct? Because α + β is in the second quadrant, so cosine is negative.cos α is √5 / 5, correct.sin(α + β) is 3/5, correct.sin α is 2√5 / 5, correct.So, plugging into the formula:cos β = (-4/5)(√5 / 5) + (3/5)(2√5 / 5)First term: (-4/5)(√5 / 5) = (-4√5)/25Second term: (3/5)(2√5 / 5) = (6√5)/25Adding them: (-4√5 + 6√5)/25 = (2√5)/25Yes, that seems correct.So, cos β = 2√5 / 25.Looking at the options, that's option A.But wait, let me think again. Is there another possibility? Because sometimes when dealing with trigonometric equations, there can be multiple solutions.Wait, in this case, we have α and β as acute angles, so β must be between 0 and 90 degrees. Therefore, cos β must be positive, which it is in this case, 2√5 / 25 is positive.Is there another possible value for cos β? Let me think.We used the cosine of a difference formula, which gave us a unique value. So, unless there's another angle that satisfies the given conditions, this should be the only solution.Wait, but let me consider if α + β could be in the first quadrant. If α + β were in the first quadrant, then cos(α + β) would be positive, 4/5. Let me see what would happen in that case.If cos(α + β) = 4/5, then cos β would be:cos β = (4/5)(√5 / 5) + (3/5)(2√5 / 5) = (4√5)/25 + (6√5)/25 = (10√5)/25 = (2√5)/5.So, cos β could be either 2√5 / 25 or 2√5 / 5, depending on whether α + β is in the first or second quadrant.But earlier, I concluded that α + β must be in the second quadrant because α is greater than 60 degrees, and β is acute, so their sum is greater than 60 degrees. But sin(α + β) = 3/5, which is approximately 0.6, which corresponds to angles around 36.87 degrees and 143.13 degrees.But wait, α is greater than 60 degrees, so α + β is greater than 60 degrees. So, the angle 36.87 degrees is less than 60 degrees, so it's not possible. Therefore, α + β must be 143.13 degrees, which is in the second quadrant. Therefore, cos(α + β) must be negative, so only the first solution is valid, which is 2√5 / 25.Wait, but let me think again. If α is greater than 60 degrees, and β is acute, say β is 30 degrees, then α + β could be 90 degrees or more. So, it's possible that α + β is in the second quadrant.But if α is, for example, 60 degrees, and β is 30 degrees, then α + β is 90 degrees. So, sin(90 degrees) is 1, but in our case, sin(α + β) is 3/5, which is less than 1. So, α + β is less than 90 degrees? Wait, no, because 3/5 is approximately 0.6, which is less than 1, so it could be in the first or second quadrant.Wait, but if α is greater than 60 degrees, and β is acute, then α + β is greater than 60 degrees. So, if α + β is greater than 60 degrees, and sin(α + β) is 3/5, which is approximately 0.6, which is greater than sin(60 degrees) which is approximately 0.866. Wait, no, 0.6 is less than 0.866. So, sin(α + β) = 0.6, which is less than sin(60 degrees). So, that would imply that α + β is less than 60 degrees? But that's a contradiction because α is already greater than 60 degrees.Wait, hold on, that doesn't make sense. If α is greater than 60 degrees, and β is positive, then α + β must be greater than 60 degrees. But sin(α + β) = 3/5 is approximately 0.6, which is less than sin(60 degrees) ≈ 0.866. So, that would imply that α + β is less than 60 degrees, which is impossible because α is already greater than 60 degrees.Wait, that can't be. So, maybe my initial assumption that α is greater than 60 degrees is wrong.Wait, let's go back. cos α = √5 / 5 ≈ 0.447. So, cos 60 degrees is 0.5, which is greater than 0.447, so α is greater than 60 degrees because cosine decreases as the angle increases. So, α is greater than 60 degrees.But if α is greater than 60 degrees, and β is acute, then α + β is greater than 60 degrees. But sin(α + β) = 3/5 ≈ 0.6, which is less than sin(60 degrees) ≈ 0.866. So, how can sin(α + β) be less than sin(60 degrees) if α + β is greater than 60 degrees?Wait, that seems contradictory. Because in the first quadrant, as the angle increases, sine increases. So, if α + β is greater than 60 degrees, sin(α + β) should be greater than sin(60 degrees). But in our case, it's less. So, that suggests that α + β is actually in the second quadrant, where sine is still positive, but cosine is negative.Wait, but in the second quadrant, sin(θ) is still positive, but cos(θ) is negative. So, if α + β is in the second quadrant, say 180 - θ, where θ is in the first quadrant, then sin(α + β) = sin(θ). So, sin(α + β) = 3/5 could correspond to an angle in the first or second quadrant. But since α + β is greater than 60 degrees, and sin(α + β) is 3/5, which is less than sin(60 degrees), that suggests that α + β is actually in the second quadrant, because in the second quadrant, the sine is still positive, but the angle is greater than 90 degrees, so the sine is decreasing after 90 degrees.Wait, let me think about this. In the first quadrant, from 0 to 90 degrees, sine increases from 0 to 1. In the second quadrant, from 90 to 180 degrees, sine decreases from 1 back to 0. So, sin(α + β) = 3/5 could correspond to an angle in the first quadrant, say θ, or in the second quadrant, 180 - θ.But since α + β is greater than 60 degrees, and sin(α + β) = 3/5 is less than sin(60 degrees) ≈ 0.866, that suggests that α + β is in the second quadrant, because in the first quadrant, sin(α + β) would have to be greater than sin(60 degrees) if α + β is greater than 60 degrees.Wait, that makes sense. So, if α + β is in the first quadrant, greater than 60 degrees, sin(α + β) should be greater than sin(60 degrees). But since it's less, α + β must be in the second quadrant, where the sine is still positive but decreasing.Therefore, α + β is in the second quadrant, so cos(α + β) is negative, which we already considered earlier.So, going back, cos β = 2√5 / 25.But wait, earlier I thought that if cos(α + β) were positive, we would get another solution, but now I see that α + β must be in the second quadrant, so cos(α + β) must be negative, so only one solution is possible.Therefore, the answer is 2√5 / 25, which is option A.But wait, let me check the options again. Option A is 2√5 / 25, and option B is 2√5 / 5. Option C says either A or B, and option D says either B or A. Wait, no, looking back:Wait, the options are:A: 2√5 / 25B: 2√5 / 5C: 2√5 / 25 or 2√5 / 5D: 2√5 / 5 or 2√5 / 25Wait, so options C and D are the same, just written in different orders. So, the correct answer is either A or B, depending on the quadrant.But from our earlier reasoning, α + β must be in the second quadrant, so only A is valid. So, the answer should be A.But wait, let me think again. If α + β were in the first quadrant, would that be possible?If α + β were in the first quadrant, then cos(α + β) would be positive, 4/5, and then cos β would be:cos β = (4/5)(√5 / 5) + (3/5)(2√5 / 5) = (4√5 + 6√5)/25 = 10√5 / 25 = 2√5 / 5.So, that's option B.But earlier, I concluded that α + β must be in the second quadrant because sin(α + β) = 3/5 is less than sin(60 degrees), which would require α + β to be in the second quadrant.But wait, let's verify this with actual angle measures.If α is greater than 60 degrees, say α = 60 degrees, then cos α = 0.5, but in our case, cos α = √5 / 5 ≈ 0.447, so α is greater than 60 degrees, say 63.43 degrees, because tan⁻¹(2) is approximately 63.43 degrees, and cos(63.43 degrees) is √5 / 5.So, α ≈ 63.43 degrees.If α ≈ 63.43 degrees, and sin(α + β) = 3/5, which is approximately 0.6, then α + β ≈ sin⁻¹(0.6) ≈ 36.87 degrees or 143.13 degrees.But α is already 63.43 degrees, so α + β cannot be 36.87 degrees because that would require β to be negative, which is impossible since β is acute.Therefore, α + β must be 143.13 degrees, which is in the second quadrant. Therefore, cos(α + β) is negative, so cos β = 2√5 / 25.Therefore, the correct answer is A.But wait, let me think again. If α is 63.43 degrees, and α + β is 143.13 degrees, then β is 143.13 - 63.43 ≈ 79.7 degrees, which is still acute, so that's valid.Alternatively, if α + β were in the first quadrant, say 36.87 degrees, but that would require β to be negative, which is impossible. So, only the second quadrant solution is valid.Therefore, cos β is 2√5 / 25, which is option A.Wait, but in the initial problem, the options are A: 2√5 / 25, B: 2√5 / 5, C: either A or B, D: either B or A.But from my reasoning, only A is valid. So, why is option C and D suggesting multiple answers? Maybe I missed something.Wait, perhaps I made a mistake in assuming that α + β must be in the second quadrant. Let me think again.If α is greater than 60 degrees, and β is acute, then α + β is greater than 60 degrees. But sin(α + β) = 3/5 is approximately 0.6, which is less than sin(60 degrees) ≈ 0.866. So, in the first quadrant, if α + β were less than 60 degrees, sin(α + β) would be less than sin(60 degrees). But since α is already greater than 60 degrees, α + β cannot be less than 60 degrees. Therefore, α + β must be in the second quadrant, where sin is still positive, but the angle is greater than 90 degrees.Therefore, only the second quadrant solution is valid, so cos β = 2√5 / 25, which is option A.But wait, let me check with actual numbers. Let me compute α and β numerically.Given cos α = √5 / 5 ≈ 0.447, so α ≈ arccos(0.447) ≈ 63.43 degrees.Given sin(α + β) = 3/5 = 0.6, so α + β ≈ arcsin(0.6) ≈ 36.87 degrees or 143.13 degrees.But α is 63.43 degrees, so α + β cannot be 36.87 degrees because that would require β to be negative, which is impossible. Therefore, α + β must be 143.13 degrees.Therefore, β = 143.13 - 63.43 ≈ 79.7 degrees.Now, let's compute cos β. cos(79.7 degrees) ≈ cos(80 degrees) ≈ 0.1736.Now, let's compute 2√5 / 25 ≈ 2*2.236 / 25 ≈ 4.472 / 25 ≈ 0.1789, which is approximately 0.1736, which is close enough considering rounding errors.Alternatively, 2√5 / 5 ≈ 4.472 / 5 ≈ 0.8944, which is much larger, corresponding to an angle of about 26.565 degrees, which would make α + β = 63.43 + 26.565 ≈ 90 degrees, but sin(90 degrees) is 1, not 3/5. So, that's not possible.Therefore, only 2√5 / 25 is valid.So, the answer is A.But wait, the options include C and D which suggest multiple answers. Maybe the problem allows for multiple solutions, but in this case, only one solution is valid because of the constraints on α and β being acute.Therefore, the correct answer is A: 2√5 / 25.

answer：First, I need to determine the number of people in the group. Let's denote this number as ( n ).The average weight of the group increases by 5.5 kg when a new person replaces someone who weighs 86 kg. The new person weighs 135.5 kg.The increase in total weight due to the new person is the difference between the new person's weight and the weight of the person being replaced:[135.5 , text{kg} - 86 , text{kg} = 49.5 , text{kg}]This increase in total weight is also equal to the product of the number of people and the increase in average weight:[n times 5.5 , text{kg} = 49.5 , text{kg}]To find ( n ), I divide the total increase in weight by the increase in average weight:[n = frac{49.5}{5.5} = 9]Therefore, there are 9 people in the group.

answer：Alright, so I have this problem here about sets. The universal set U is given as {1, 2, 3, 4, 5, 6}, and there's another set M which is {1, 2, 4}. I need to find the complement of M in U, which is denoted as ∁ₐM. The options are A: U, B: {1, 3, 5}, C: {3, 5, 6}, and D: {2, 4, 6}.Okay, first, I need to remember what a complement of a set is. From what I recall, the complement of a set M with respect to the universal set U is the set of all elements in U that are not in M. So, basically, it's like taking everything that's in U but not in M.Let me write that down to make it clearer. The complement of M, ∁ₐM, should include all elements that are in U but not in M. So, U is {1, 2, 3, 4, 5, 6}, and M is {1, 2, 4}. If I remove the elements that are in M from U, what's left should be the complement.Let me list out the elements of U: 1, 2, 3, 4, 5, 6.Now, the elements in M are 1, 2, 4. So, if I take away 1, 2, and 4 from U, what's left? That would be 3, 5, and 6.So, ∁ₐM should be {3, 5, 6}.Looking at the options, option C is {3, 5, 6}, which matches what I just figured out. So, I think the answer is C.Wait, let me double-check to make sure I didn't make a mistake. Sometimes, I get confused with complements and other set operations.So, U is {1, 2, 3, 4, 5, 6}, and M is {1, 2, 4}. The complement of M in U would indeed be all elements in U not in M. So, removing 1, 2, and 4 from U leaves 3, 5, and 6. Yep, that seems right.I think I'm confident now that option C is the correct answer.

answer：First, I recognize that the pitcher is three-fourths full, which means it contains (frac{3}{4}) of its total capacity in orange juice.Next, this amount of juice is distributed equally into 8 cups. To find out how much juice each cup receives, I divide the total juice by the number of cups:[frac{frac{3}{4}}{8} = frac{3}{32}]This fraction represents the portion of the pitcher's total capacity that each cup receives. To convert this fraction into a percentage, I multiply by 100:[frac{3}{32} times 100 = 9.375%]Since the question asks for the percentage and the options provided are whole numbers, I round 9.375% to the nearest whole number, which is 9%.Therefore, each cup received approximately 9% of the total capacity of the pitcher.

answer：First, I need to determine the scale of the map by comparing the actual distance between the two mountains to the distance shown on the map. The actual distance is 136 km, and the map distance is 312 inches.Next, I'll convert the actual distance Ram is from the base of the mountain (14.82 km) into inches to maintain consistent units. Knowing that 1 kilometer is approximately 39370.1 inches, I can calculate Ram's distance in inches.After obtaining Ram's distance in inches, I'll use the previously determined scale to find out how far Ram's camp is from the base of the mountain on the map. This involves dividing Ram's distance in inches by the scale to get the map distance in inches.Finally, I'll present the calculated map distance as the answer.