💡Okay, so I have this vector problem here. Let me read it again to make sure I understand what's being asked. It says:"Let vectors a and b be two vectors with |a| = 1 and |b| = 2. Given that a + b is perpendicular to a, find the angle between a and b. The options are A: 30°, B: 60°, C: 120°, D: 150°."Alright, so I need to find the angle between vectors a and b given their magnitudes and the condition that a + b is perpendicular to a. Hmm, okay. Let's break this down step by step.First, I remember that if two vectors are perpendicular, their dot product is zero. So, if a + b is perpendicular to a, then the dot product of (a + b) and a should be zero. That gives me an equation to work with.Let me write that down:(a + b) · a = 0Now, I can expand this dot product using the distributive property of dot products. So, that would be:a · a + b · a = 0I know that the dot product of a vector with itself is equal to the square of its magnitude. So, a · a is |a|², and similarly, b · a is the same as a · b because the dot product is commutative. Given that |a| = 1, |a|² is 1², which is 1. So, substituting that in:1 + a · b = 0Now, I need to find a · b. Let me rearrange the equation:a · b = -1But I also know that the dot product of two vectors can be expressed in terms of their magnitudes and the cosine of the angle between them. The formula is:a · b = |a| |b| cosθWhere θ is the angle between vectors a and b. We know |a| is 1 and |b| is 2, so substituting those values in:a · b = 1 * 2 * cosθ = 2 cosθBut earlier, we found that a · b = -1. So, setting these equal:2 cosθ = -1Now, solving for cosθ:cosθ = -1/2Okay, so cosθ is -1/2. Now, I need to find the angle θ where the cosine is -1/2. I remember from the unit circle that cosine is -1/2 at 120° and 240°, but since angles between vectors are always between 0° and 180°, 240° is out of the question. So, θ must be 120°.Let me double-check my steps to make sure I didn't make any mistakes. 1. I started with the condition that (a + b) is perpendicular to a, so their dot product is zero.2. Expanded the dot product to get a · a + a · b = 0.3. Substituted |a|² = 1 into the equation.4. Found that a · b = -1.5. Expressed a · b in terms of magnitudes and cosine of the angle.6. Solved for cosθ and found it to be -1/2.7. Concluded that θ is 120° since that's the angle between 0° and 180° where cosine is -1/2.Everything seems to check out. I think I'm confident that 120° is the correct answer. Just to visualize, if vector a has a magnitude of 1 and vector b has a magnitude of 2, and they form a 120° angle, then when you add them together, the resultant vector a + b is perpendicular to a. That makes sense because the components would balance out in such a way that the resultant vector is at a right angle to a. I don't see any errors in my reasoning, so I think I'm good. The answer should be 120°, which is option C.

💡Okay, so I've got this problem here about fish in a tank. Let me try to understand it step by step. The problem says there are many fish in the tank, and one third of them are blue. Then, it mentions that half of the blue fish have spots. Finally, it tells me that there are 10 blue, spotted fish, and I need to find out how many fish there are in total in the tank.Alright, let's break this down. First, it's talking about fractions of the total fish population. So, if one third of the fish are blue, that means if I let the total number of fish be, say, F, then the number of blue fish would be F divided by 3, or F/3.Next, it says that half of the blue fish have spots. So, if I have F/3 blue fish, then half of that number would be the number of blue, spotted fish. That would be (F/3) divided by 2, which simplifies to F/6.But wait, the problem tells me that there are 10 blue, spotted fish. So, according to my calculation, F/6 equals 10. To find F, I can set up the equation F/6 = 10 and solve for F.Multiplying both sides of the equation by 6 gives me F = 60. So, does that mean there are 60 fish in total in the tank? Let me double-check to make sure I didn't make any mistakes.If there are 60 fish in total, then one third of them are blue. One third of 60 is 20, so there are 20 blue fish. Half of those blue fish have spots, which would be 10 blue, spotted fish. That matches the information given in the problem, so it seems like my answer is correct.But just to be thorough, let me go through it again. Starting with the total number of fish, F. One third are blue, so blue fish = F/3. Half of those blue fish have spots, so spotted blue fish = (F/3)/2 = F/6. Given that spotted blue fish = 10, so F/6 = 10, which means F = 60.Yep, that all adds up. I think I've got it right. There are 60 fish in the tank.

💡Okay, so I have this math problem here: Calculate (1005^2 - 995^2 - 1007^2 + 993^2) without using a calculator. Hmm, that looks a bit intimidating at first glance because of all those large numbers, but maybe there's a way to simplify it.I remember something about the difference of squares from algebra. The formula is (a^2 - b^2 = (a + b)(a - b)). Maybe I can apply that here to make the calculations easier. Let me see how I can pair these terms.Looking at the expression, I notice that there are two positive squares and two negative squares. Maybe I can group them in a way that allows me to use the difference of squares formula. Let me try pairing (1005^2 - 995^2) first.Applying the difference of squares to (1005^2 - 995^2), I get:[(1005 + 995)(1005 - 995)]Calculating the sums and differences inside the parentheses:[(2000)(10) = 20000]Okay, that worked out nicely. Now, let's look at the other pair: (-1007^2 + 993^2). I can rewrite this as (993^2 - 1007^2) to make it fit the difference of squares formula.Applying the formula:[(993 + 1007)(993 - 1007)]Calculating the sums and differences:[(2000)(-14) = -28000]Wait, that's negative because (993 - 1007) is negative. So, putting it all together, the original expression becomes:[20000 - 28000]Which simplifies to:[-8000]So, the final result should be (-8000). Let me just double-check my steps to make sure I didn't make a mistake.First, I used the difference of squares on (1005^2 - 995^2) and got 20000. Then, I applied the same formula to (993^2 - 1007^2) and got -28000. Subtracting these gives -8000. That seems correct. I don't think I made any calculation errors because the numbers worked out neatly. Yeah, I'm confident with this answer.

💡Alright, so I've got this problem here about a company buying new equipment to save energy and reduce emissions. The equipment costs 90,000, and it generates 110,000 in annual revenue. But there are operating costs involved too. In the first year, the operating cost is 20,000, and each subsequent year, this cost increases by another 20,000. The question is asking after how many years the annual average profit will be maximized. The options are 3, 4, 5, or 6 years.Okay, let me break this down step by step. First, I need to understand what exactly is being asked. They want the number of years, n, where the average profit is the highest. So, average profit would be total profit divided by the number of years, right?Let me think about how to model this. The company spends 90,000 initially, and then each year they have operating costs that increase by 20,000. So, the first year's operating cost is 20,000, the second year is 40,000, the third year is 60,000, and so on. That seems like an arithmetic sequence where each term increases by 20,000.On the revenue side, they make 110,000 each year. So, the revenue is constant every year, but the operating costs are increasing. The total profit each year would be revenue minus operating costs, but we also have to consider the initial cost of the equipment.Wait, so the initial cost is a one-time expense, right? So, that's like a fixed cost, and then each year's operating cost is variable and increasing. So, to find the total profit over n years, I need to subtract both the initial cost and the sum of all operating costs from the total revenue over n years.Let me write this out mathematically. Let's denote:- Initial cost: C = 90,000- Annual revenue: R = 110,000- Operating cost in the first year: O₁ = 20,000- Increase in operating cost each year: d = 20,000So, the operating cost in year k is O_k = O₁ + (k - 1)d. That simplifies to O_k = 20,000 + (k - 1)*20,000 = 20,000k.Wait, hold on, that would mean in the first year, k=1, O₁=20,000*1=20,000, which is correct. Second year, k=2, O₂=40,000, which matches the problem statement. So, that seems right.Now, the total operating cost over n years is the sum of this arithmetic sequence. The formula for the sum of the first n terms of an arithmetic sequence is S_n = n/2*(2a + (n - 1)d), where a is the first term.Plugging in the values, S_n = n/2*(2*20,000 + (n - 1)*20,000) = n/2*(40,000 + 20,000n - 20,000) = n/2*(20,000n + 20,000) = n/2*(20,000(n + 1)) = 10,000n(n + 1).So, total operating cost over n years is 10,000n(n + 1).Total revenue over n years is R*n = 110,000n.Total profit over n years, P_total, is total revenue minus initial cost minus total operating cost:P_total = 110,000n - 90,000 - 10,000n(n + 1).Simplify that:P_total = 110,000n - 90,000 - 10,000n² - 10,000nCombine like terms:P_total = (110,000n - 10,000n) - 10,000n² - 90,000P_total = 100,000n - 10,000n² - 90,000We can factor out 10,000 to make it simpler:P_total = 10,000*(10n - n² - 9)So, P_total = -10,000n² + 100,000n - 90,000Now, the annual average profit, P_avg, is total profit divided by n:P_avg = P_total / n = (-10,000n² + 100,000n - 90,000) / nSimplify that:P_avg = -10,000n + 100,000 - 90,000/nSo, P_avg = -10,000n + 100,000 - 90,000/nWe need to find the value of n that maximizes P_avg.Hmm, this is a function of n, and we need to find its maximum. Since n is a positive integer, we can treat this as a function of a real variable and find its maximum, then check the integer values around it.Let me denote f(n) = -10,000n + 100,000 - 90,000/nTo find the maximum, we can take the derivative of f(n) with respect to n and set it equal to zero.f'(n) = derivative of (-10,000n) + derivative of 100,000 + derivative of (-90,000/n)f'(n) = -10,000 + 0 + (90,000)/n²Set f'(n) = 0:-10,000 + 90,000/n² = 0Move the 10,000 to the other side:90,000/n² = 10,000Multiply both sides by n²:90,000 = 10,000n²Divide both sides by 10,000:9 = n²Take square root:n = 3Since n must be positive, n = 3.So, the maximum average profit occurs at n = 3 years.But wait, let me double-check this because sometimes when dealing with discrete variables, the maximum might not exactly be at the integer value found by calculus.Let me compute P_avg for n = 2, 3, 4, and maybe 5 to see.First, n = 2:P_total = -10,000*(2)^2 + 100,000*2 - 90,000 = -40,000 + 200,000 - 90,000 = 70,000P_avg = 70,000 / 2 = 35,000n = 3:P_total = -10,000*(3)^2 + 100,000*3 - 90,000 = -90,000 + 300,000 - 90,000 = 120,000P_avg = 120,000 / 3 = 40,000n = 4:P_total = -10,000*(4)^2 + 100,000*4 - 90,000 = -160,000 + 400,000 - 90,000 = 150,000P_avg = 150,000 / 4 = 37,500n = 5:P_total = -10,000*(5)^2 + 100,000*5 - 90,000 = -250,000 + 500,000 - 90,000 = 160,000P_avg = 160,000 / 5 = 32,000n = 6:P_total = -10,000*(6)^2 + 100,000*6 - 90,000 = -360,000 + 600,000 - 90,000 = 150,000P_avg = 150,000 / 6 = 25,000So, looking at these values:n=2: 35,000n=3: 40,000n=4: 37,500n=5: 32,000n=6: 25,000So, the maximum average profit is indeed at n=3, which is 40,000.Therefore, the answer should be D: 3.But wait, just to be thorough, let me check n=1 as well.n=1:P_total = -10,000*(1)^2 + 100,000*1 - 90,000 = -10,000 + 100,000 - 90,000 = 0P_avg = 0 / 1 = 0So, n=1 gives zero average profit, which makes sense because the initial cost is 90,000, and the first year's revenue is 110,000, but operating cost is 20,000, so net profit is 110,000 - 20,000 - 90,000 = 0.So, the average profit starts at zero, increases to 40,000 at n=3, then decreases afterward.Therefore, n=3 is indeed the year where the annual average profit is maximized.I think that makes sense. The operating costs are increasing each year, so after a certain point, the increasing costs start to eat into the profits more, reducing the average profit. So, the optimal point is somewhere in the middle, which in this case is 3 years.Yeah, I feel confident now that D: 3 is the correct answer.**Final Answer**boxed{D}

💡Okay, so I have this equation: (1 + i)z = 3 - i, and I need to find the modulus of z, which is |z|. Hmm, I remember that modulus of a complex number is like its distance from the origin in the complex plane. So if z is a complex number, say z = a + bi, then |z| is sqrt(a² + b²). Alright, so first, I need to find z. The equation is (1 + i)z = 3 - i. To solve for z, I should divide both sides by (1 + i). That makes sense because if I have something multiplied by z, dividing by that something should give me z. So, z = (3 - i)/(1 + i). But wait, dividing complex numbers can be tricky. I think I need to multiply the numerator and the denominator by the conjugate of the denominator to simplify it. The conjugate of (1 + i) is (1 - i). So, I'll do that:z = [(3 - i)(1 - i)] / [(1 + i)(1 - i)]Let me compute the denominator first. (1 + i)(1 - i) is a difference of squares, so it should be 1² - (i)², which is 1 - (-1) because i² is -1. So, 1 - (-1) is 1 + 1, which is 2. Okay, the denominator simplifies to 2.Now, the numerator: (3 - i)(1 - i). I'll use the distributive property (FOIL) to multiply these out.First, 3*1 = 3.Outer, 3*(-i) = -3i.Inner, (-i)*1 = -i.Last, (-i)*(-i) = i², which is -1.So, adding all these together: 3 - 3i - i - 1. Combine like terms: 3 - 1 is 2, and -3i - i is -4i. So, the numerator simplifies to 2 - 4i.Putting it all together, z = (2 - 4i)/2. I can divide both terms in the numerator by 2: 2/2 = 1, and -4i/2 = -2i. So, z = 1 - 2i.Now, to find |z|, which is the modulus of z. Since z = 1 - 2i, the modulus is sqrt(1² + (-2)²) = sqrt(1 + 4) = sqrt(5). Let me double-check my steps to make sure I didn't make a mistake. Starting from (1 + i)z = 3 - i, I divided both sides by (1 + i) to get z. Then I multiplied numerator and denominator by (1 - i), which is the conjugate. The denominator became 2, and the numerator became 2 - 4i. Dividing by 2 gave me z = 1 - 2i. Then, calculating the modulus as sqrt(1 + 4) = sqrt(5). That seems right.I think the answer is sqrt(5), which is option B.

💡Alright, so I've got this problem about air pollution index related to time, and it's described by this function f(x) = |log₂₅(x + 1) - a| + 2a + 1. The variable x is the time in hours, ranging from 0 to 24, and a is some adjustment parameter between 0 and 1. There are two parts to this problem.First, when a is 1/2, I need to find the time of day when the air pollution index is the lowest. Okay, so I need to find the minimum value of f(x) when a is 1/2. Since it's an absolute value function, the minimum should occur where the expression inside the absolute value is zero. That makes sense because the absolute value function V-shaped, so the vertex is the minimum point.So, if I set log₂₅(x + 1) - a equal to zero, that should give me the x where the minimum occurs. Plugging in a = 1/2, we have log₂₅(x + 1) - 1/2 = 0. Solving for x, log₂₅(x + 1) = 1/2. To get rid of the logarithm, I can rewrite this as 25^(1/2) = x + 1. 25^(1/2) is 5, so 5 = x + 1, which means x = 4. So, the pollution index is lowest at the 4th hour. That seems straightforward.Now, moving on to the second part. The maximum value of f(x) each day is considered the daily air pollution index, and we need to ensure that this doesn't exceed 3. So, we need to find the range of a such that the maximum of f(x) is less than or equal to 3.First, let's understand the function f(x) = |log₂₅(x + 1) - a| + 2a + 1. Since it's an absolute value function, it will have a V-shape. The expression inside the absolute value, log₂₅(x + 1) - a, will determine where the vertex of the V is. The vertex occurs where log₂₅(x + 1) = a, which is similar to part (1). Solving for x, we get x = 25^a - 1. So, the function f(x) will decrease until x = 25^a - 1 and then increase after that point.Therefore, the maximum value of f(x) will occur at one of the endpoints of the interval, either at x = 0 or x = 24, because the function is decreasing before the vertex and increasing after. So, we need to evaluate f(x) at both endpoints and ensure that both are less than or equal to 3.Let's compute f(0). Plugging x = 0 into f(x), we get |log₂₅(0 + 1) - a| + 2a + 1. log₂₅(1) is 0, so this simplifies to |0 - a| + 2a + 1, which is | -a | + 2a + 1. Since a is positive, this is a + 2a + 1 = 3a + 1.Next, compute f(24). Plugging x = 24 into f(x), we get |log₂₅(24 + 1) - a| + 2a + 1. log₂₅(25) is 1, so this becomes |1 - a| + 2a + 1. Since a is between 0 and 1, 1 - a is positive, so the absolute value is just 1 - a. Therefore, f(24) = (1 - a) + 2a + 1 = 1 - a + 2a + 1 = a + 2.So, we have two expressions for the maximum value: f(0) = 3a + 1 and f(24) = a + 2. We need both of these to be less than or equal to 3.Let's set up the inequalities:1. 3a + 1 ≤ 32. a + 2 ≤ 3Solving the first inequality: 3a + 1 ≤ 3. Subtract 1 from both sides: 3a ≤ 2. Divide by 3: a ≤ 2/3.Solving the second inequality: a + 2 ≤ 3. Subtract 2: a ≤ 1.But we already know that a is in (0, 1), so combining these results, a must be less than or equal to 2/3. So, the range for a is (0, 2/3].Wait, let me double-check. If a is 2/3, then f(0) = 3*(2/3) + 1 = 2 + 1 = 3, which is exactly the limit. And f(24) = (2/3) + 2 = 8/3 ≈ 2.666..., which is less than 3. So, that works.If a is greater than 2/3, say a = 0.7, then f(0) = 3*0.7 + 1 = 2.1 + 1 = 3.1, which exceeds 3. So, that's not allowed. Hence, a must be at most 2/3.Therefore, the adjustment parameter a should be controlled within the range (0, 2/3].**Final Answer**(1) The air pollution index is lowest at the boxed{4}th hour of the day.(2) The adjustment parameter ( a ) should be controlled within the range boxed{left(0, dfrac{2}{3}right]}.