💡Okay, so I have this problem here about vectors and projections. Let me try to understand it step by step. We're given two vectors, OA and OB, with coordinates OA = (1, 4) and OB = (-3, 1). The problem says that the projections of OA and OB onto the direction vector of line l have equal lengths. Also, it mentions that the angle of inclination of line l is obtuse, and we need to find the slope of line l.Hmm, projections of vectors onto another vector... I remember that the projection of a vector a onto another vector u is given by the dot product of a and u divided by the magnitude of u. So, mathematically, that would be (a · u) / |u|. Since the projections have equal lengths, their absolute values should be equal. So, if I denote the direction vector of line l as u = (m, n), then the projection of OA onto u is (OA · u) / |u|, and similarly for OB. Since these projections have equal lengths, their absolute values are equal. That gives me the equation:| (OA · u) / |u| | = | (OB · u) / |u| |Since |u| is a positive scalar, it cancels out from both sides, leaving me with:| OA · u | = | OB · u |Calculating the dot products:OA · u = (1)(m) + (4)(n) = m + 4nOB · u = (-3)(m) + (1)(n) = -3m + nSo, setting their absolute values equal:| m + 4n | = | -3m + n |This equation implies that either m + 4n = -3m + n or m + 4n = -(-3m + n), which simplifies to m + 4n = 3m - n.Let me write that down:Case 1: m + 4n = -3m + nCase 2: m + 4n = 3m - nLet me solve both cases.Starting with Case 1:m + 4n = -3m + nBring all terms to one side:m + 4n + 3m - n = 0Combine like terms:4m + 3n = 0So, 4m + 3n = 0Which can be rewritten as:4m = -3nDivide both sides by m (assuming m ≠ 0):4 = -3(n/m)So, n/m = -4/3Which means the slope of line l is n/m = -4/3Now, moving to Case 2:m + 4n = 3m - nBring all terms to one side:m + 4n - 3m + n = 0Combine like terms:-2m + 5n = 0So, -2m + 5n = 0Which can be rewritten as:5n = 2mDivide both sides by m (assuming m ≠ 0):5(n/m) = 2So, n/m = 2/5Which would mean the slope is 2/5.But wait, the problem says that the angle of inclination of line l is obtuse. An obtuse angle means that the angle is greater than 90 degrees but less than 180 degrees. The slope of a line is related to the tangent of the angle of inclination. If the angle is obtuse, the slope should be negative because the tangent of an angle greater than 90 degrees is negative.So, in Case 1, the slope is -4/3, which is negative, and in Case 2, the slope is 2/5, which is positive. Since we need an obtuse angle, we should choose the negative slope.Therefore, the slope of line l is -4/3.Let me just double-check my steps to make sure I didn't make any mistakes.1. I set the direction vector as u = (m, n).2. Calculated the dot products correctly: OA · u = m + 4n and OB · u = -3m + n.3. Set the absolute values equal, leading to two cases.4. Solved both cases, getting slopes of -4/3 and 2/5.5. Considered the angle of inclination being obtuse, which requires a negative slope, so selected -4/3.Everything seems to check out. I think that's the correct answer.**Final Answer**The slope of line ( l ) is boxed{-dfrac{4}{3}}.

💡Okay, so I need to figure out the arithmetic mean of the first n positive integers and the arithmetic mean of the squares of the first n positive integers. Hmm, let me break this down step by step.First, let me recall what the arithmetic mean is. It's just the sum of all the numbers divided by how many numbers there are. So, for the first part, I need the sum of the first n positive integers and then divide that by n. For the second part, I need the sum of the squares of the first n positive integers and then divide that by n as well.Starting with the first part: the arithmetic mean of the first n positive integers. I remember there's a formula for the sum of the first n positive integers. Let me see... I think it's something like n times (n + 1) divided by 2. Yeah, that sounds right. So, the sum S of the first n integers is S = n(n + 1)/2.Now, to find the arithmetic mean, I need to divide this sum by n. So, the mean would be (n(n + 1)/2) divided by n. Let me write that out: (n(n + 1)/2) / n. The n in the numerator and the denominator should cancel out, right? So, that simplifies to (n + 1)/2. Okay, that seems straightforward.Now, moving on to the second part: the arithmetic mean of the squares of the first n positive integers. I think there's a formula for the sum of squares as well. Let me try to remember... I believe it's n(n + 1)(2n + 1)/6. Yeah, that rings a bell. So, the sum of the squares S_sq is S_sq = n(n + 1)(2n + 1)/6.To find the arithmetic mean, I need to divide this sum by n. So, the mean would be (n(n + 1)(2n + 1)/6) divided by n. Let me write that: (n(n + 1)(2n + 1)/6) / n. Again, the n in the numerator and denominator should cancel out, leaving me with (n + 1)(2n + 1)/6.Let me double-check these results to make sure I didn't make a mistake. For the first part, if n = 1, the mean should be 1. Plugging into (1 + 1)/2 = 1, which is correct. If n = 2, the mean should be (1 + 2)/2 = 1.5. Plugging into (2 + 1)/2 = 1.5, that's also correct.For the second part, if n = 1, the mean should be 1^2 = 1. Plugging into (1 + 1)(2*1 + 1)/6 = (2)(3)/6 = 6/6 = 1, which is correct. If n = 2, the mean should be (1^2 + 2^2)/2 = (1 + 4)/2 = 5/2 = 2.5. Plugging into (2 + 1)(2*2 + 1)/6 = (3)(5)/6 = 15/6 = 2.5, which is also correct.So, both formulas seem to work for small values of n. I think I did this right. Therefore, the arithmetic mean of the first n positive integers is (n + 1)/2, and the arithmetic mean of their squares is (n + 1)(2n + 1)/6.Looking back at the options provided:(A) (n+1)/2 and (n³ + n²)/6 (B) n/2 and n²(n+1)/6 (C) (n+1)/2 and (n+1)(2n+1)/6 (D) (n-1)/2 and (n-1)(2n-1)/6Comparing my results with the options, option C matches exactly. So, I think the correct answer is option C.

💡First, I need to calculate the total cost of the items in Alice's cart, excluding the broccoli since the quantity is unknown.The cost of the chicken is 1.5 pounds at 6.00 per pound:1.5 pounds * 6.00/pound = 9.00The cost of the lettuce is 3.00.The cost of the cherry tomatoes is 2.50.The cost of the sweet potatoes is 4 at 0.75 each:4 * 0.75 = 3.00The cost of the Brussel sprouts is 2.50.Adding up these costs:9.00 (chicken) + 3.00 (lettuce) + 2.50 (cherry tomatoes) + 3.00 (sweet potatoes) + 2.50 (Brussel sprouts) = 20.00Alice needs to spend a minimum of 35.00 for free delivery. She currently has 20.00 worth of items, so she needs:35.00 - 20.00 = 15.00 moreShe needs to spend 11 more to reach the free delivery threshold, which means she already has:15.00 - 11.00 = 4.00 worth of broccoli in her cart.Since each head of broccoli costs 2.00, the number of heads of broccoli she has is:4.00 / 2.00 per head = 2 heads of broccoli

💡Okay, so I have this problem about Pascal's Triangle. It says, "In a row of Pascal's triangle that begins with 1 and then 15, what is the fifth number?" Hmm, I need to figure out what row this is and then find the fifth number in that row.First, I remember that each row in Pascal's Triangle starts and ends with 1, and the numbers in between are combinations. The first number is always 1, and the second number is the row number itself. So if the row starts with 1 and then 15, that must mean it's the 15th row, right? Because the second number in the nth row is n.So, the row we're looking at is the 15th row. Now, I need to find the fifth number in this row. I know that the numbers in Pascal's Triangle correspond to binomial coefficients. The kth number in the nth row is given by the combination formula: C(n, k-1) because we start counting from 0.Wait, let me make sure I got that right. If the first number is C(n, 0), then the second is C(n, 1), and so on. So, the fifth number would be C(n, 4) because we start counting from 0. Since n is 15, the fifth number is C(15, 4).Okay, now I need to calculate C(15, 4). The formula for combinations is C(n, k) = n! / (k!(n - k)!). So, plugging in the numbers, C(15, 4) = 15! / (4! * (15 - 4)!).Simplifying that, 15 - 4 is 11, so it becomes 15! / (4! * 11!). I know that 15! is a huge number, but maybe I can simplify it before calculating. Let's see, 15! is 15 × 14 × 13 × 12 × 11!, right? So, I can cancel out the 11! in the numerator and denominator.That leaves me with (15 × 14 × 13 × 12) / (4!). Now, 4! is 4 × 3 × 2 × 1, which is 24. So, I have (15 × 14 × 13 × 12) / 24.Let me compute the numerator first: 15 × 14 is 210, 210 × 13 is 2730, and 2730 × 12 is 32760. So, the numerator is 32760.Now, divide that by 24: 32760 ÷ 24. Let me do that step by step. 24 × 1000 is 24,000, which is less than 32,760. The difference is 8,760. 24 × 300 is 7,200, so subtracting that from 8,760 leaves 1,560. 24 × 65 is 1,560. So, adding up, 1000 + 300 + 65 is 1,365.So, the fifth number in the 15th row of Pascal's Triangle is 1,365.Wait, let me double-check my calculations to make sure I didn't make a mistake. Starting with 15 × 14 × 13 × 12, that's 15 × 14 = 210, 210 × 13 = 2730, 2730 × 12 = 32,760. Divided by 24, which is 32,760 ÷ 24. 24 × 1,000 = 24,000, subtract that from 32,760, you get 8,760. 24 × 300 = 7,200, subtract that from 8,760, you get 1,560. 24 × 65 = 1,560. So, 1,000 + 300 + 65 = 1,365. Yeah, that seems right.I think I've got it. The fifth number is 1,365.

💡First, I recognize that the cosine function is even, meaning that (cos(-x) = cos(x)). This allows me to rewrite (cos(-750^circ)) as (cos(750^circ)).Next, I need to find an angle coterminal with (750^circ) that lies within the standard range of (0^circ) to (360^circ). To do this, I divide (750^circ) by (360^circ) to find how many full rotations there are and what the remainder is.Calculating the division:[750 div 360 = 2 text{ with a remainder of } 30]This means that (750^circ) is coterminal with (30^circ).Now, I recall the value of (cos(30^circ)), which is (frac{sqrt{3}}{2}).Therefore, (cos(-750^circ) = cos(750^circ) = cos(30^circ) = frac{sqrt{3}}{2}).

💡Okay, so I need to find the point that's exactly halfway between the two points (2,9) and (8,3). Hmm, I remember from school that there's a formula for finding the midpoint between two points. Let me try to recall how that works.First, I think the midpoint is like the average of the two points. So, if I have two points, say (x1, y1) and (x2, y2), the midpoint should be somewhere in the middle of them. I believe the formula involves taking the average of the x-coordinates and the average of the y-coordinates separately. So, the midpoint (x, y) would be:x = (x1 + x2) / 2y = (y1 + y2) / 2Let me check if that makes sense. If I have two points on a straight line, the midpoint should be the average of their positions. Yeah, that sounds right. So, applying this to the points given, which are (2,9) and (8,3).Let me write down the coordinates:Point A: (2, 9)Point B: (8, 3)So, x1 is 2, y1 is 9, x2 is 8, and y2 is 3.Now, plugging these into the midpoint formula:x = (2 + 8) / 2y = (9 + 3) / 2Let me calculate the x-coordinate first. 2 plus 8 is 10, and then divided by 2 is 5. So, x is 5.Now, the y-coordinate. 9 plus 3 is 12, and then divided by 2 is 6. So, y is 6.Therefore, the midpoint should be at (5, 6). Wait, let me visualize this to make sure. If I plot the points (2,9) and (8,3) on a graph, (2,9) is somewhere in the upper left, and (8,3) is more to the right and lower. The midpoint should be somewhere in between. If I move from (2,9) halfway towards (8,3), I should end up at (5,6). That seems reasonable.Let me double-check my calculations. For the x-coordinate: 2 plus 8 is definitely 10, and half of 10 is 5. For the y-coordinate: 9 plus 3 is 12, and half of 12 is 6. Yep, that looks correct.I think I got it right. The midpoint between (2,9) and (8,3) is (5,6). I don't see any mistakes in my reasoning or calculations. It all adds up.