💡To find the projection vector of (overrightarrow{b}) onto (overrightarrow{a}), we can use the formula for the projection of one vector onto another. The formula is:[text{proj}_{overrightarrow{a}} overrightarrow{b} = left( frac{overrightarrow{a} cdot overrightarrow{b}}{|overrightarrow{a}|^2} right) overrightarrow{a}]First, we need to compute the dot product of (overrightarrow{a}) and (overrightarrow{b}):[overrightarrow{a} cdot overrightarrow{b} = (0)(-2) + (1)(-8) = 0 - 8 = -8]Next, we calculate the magnitude of (overrightarrow{a}):[|overrightarrow{a}| = sqrt{0^2 + 1^2} = sqrt{0 + 1} = 1]Now, we can find the scalar projection factor:[frac{overrightarrow{a} cdot overrightarrow{b}}{|overrightarrow{a}|^2} = frac{-8}{1^2} = -8]Finally, we multiply this scalar by the vector (overrightarrow{a}) to get the projection vector:[text{proj}_{overrightarrow{a}} overrightarrow{b} = -8 times (0, 1) = (0, -8)]

💡Okay, so I have this problem about a sailboat and the force of the wind on its sail. The formula given is ( F = frac{C S rho (v_0 - v)^2}{2} ), where ( C ) is the aerodynamic force coefficient, ( S ) is the sail area, ( rho ) is the air density, ( v_0 ) is the wind speed, and ( v ) is the speed of the sailboat. The question is asking for the speed ( v ) of the sailboat when the instantaneous power of the wind reaches its maximum value.First, I need to understand what power means in this context. Power is generally defined as the rate at which work is done, or equivalently, the product of force and velocity. So, in this case, the power ( N ) would be the force ( F ) multiplied by the velocity ( v ) of the sailboat. That makes sense because the faster the sailboat moves, the more work it can do, but there's a balance because the force from the wind depends on the relative speed between the wind and the sailboat.So, let's write that out:[ N = F cdot v ]Substituting the given formula for ( F ):[ N = frac{C S rho (v_0 - v)^2}{2} cdot v ]Simplifying that:[ N = frac{C S rho}{2} cdot v (v_0 - v)^2 ]Now, I need to find the value of ( v ) that maximizes ( N ). To do this, I should take the derivative of ( N ) with respect to ( v ) and set it equal to zero. This will give me the critical points, and then I can determine which one gives the maximum power.Let's denote ( N ) as a function of ( v ):[ N(v) = frac{C S rho}{2} cdot v (v_0 - v)^2 ]To find the maximum, take the derivative ( N'(v) ):First, expand the expression inside the function to make differentiation easier:[ N(v) = frac{C S rho}{2} cdot v (v_0^2 - 2 v_0 v + v^2) ][ N(v) = frac{C S rho}{2} cdot (v_0^2 v - 2 v_0 v^2 + v^3) ]Now, take the derivative with respect to ( v ):[ N'(v) = frac{C S rho}{2} cdot (v_0^2 - 4 v_0 v + 3 v^2) ]Set the derivative equal to zero to find the critical points:[ frac{C S rho}{2} cdot (v_0^2 - 4 v_0 v + 3 v^2) = 0 ]Since ( frac{C S rho}{2} ) is a positive constant, we can ignore it for the purpose of solving the equation:[ v_0^2 - 4 v_0 v + 3 v^2 = 0 ]This is a quadratic equation in terms of ( v ). Let's rewrite it:[ 3 v^2 - 4 v_0 v + v_0^2 = 0 ]Now, solve for ( v ) using the quadratic formula:[ v = frac{4 v_0 pm sqrt{(4 v_0)^2 - 4 cdot 3 cdot v_0^2}}{2 cdot 3} ][ v = frac{4 v_0 pm sqrt{16 v_0^2 - 12 v_0^2}}{6} ][ v = frac{4 v_0 pm sqrt{4 v_0^2}}{6} ][ v = frac{4 v_0 pm 2 v_0}{6} ]This gives two solutions:1. ( v = frac{4 v_0 + 2 v_0}{6} = frac{6 v_0}{6} = v_0 )2. ( v = frac{4 v_0 - 2 v_0}{6} = frac{2 v_0}{6} = frac{v_0}{3} )Now, we need to interpret these solutions. If ( v = v_0 ), that would mean the sailboat is moving at the same speed as the wind. In this case, the relative speed between the wind and the sailboat is zero, so the force ( F ) would also be zero, which means the power ( N ) would be zero. Therefore, this solution doesn't make sense in the context of maximizing power.The other solution is ( v = frac{v_0}{3} ). This seems more reasonable because it's less than the wind speed, allowing for a positive relative speed and thus a positive force and power.To confirm that this is indeed a maximum, we can check the second derivative or analyze the behavior of the function. However, given the physical context, it's reasonable to conclude that ( v = frac{v_0}{3} ) is the speed at which the power is maximized.So, after going through the calculations and interpreting the results, the speed of the sailboat when the instantaneous power of the wind is maximized is ( frac{v_0}{3} ).

💡Okay, so I'm trying to figure out which department has more employees based on the number of messages sent and received. Let me start by understanding the problem.We have two departments: Android and iOS. Each developer in the Android department sends 7 messages and receives 15. Each developer in the iOS department sends 15 messages and receives 9. Messages can be sent within the department or to the other department.Hmm, so I need to find out which department has more employees. Let me denote the number of employees in the Android department as 'n' and in the iOS department as 'm'.First, let's think about the total number of messages sent. For the Android department, each employee sends 7 messages, so the total messages sent from Android would be 7n. Similarly, for iOS, each employee sends 15 messages, so the total messages sent from iOS would be 15m. Therefore, the total messages sent overall would be 7n + 15m.Now, for the messages received. Each Android employee receives 15 messages, so the total messages received by Android would be 15n. Each iOS employee receives 9 messages, so the total messages received by iOS would be 9m. Therefore, the total messages received overall would be 15n + 9m.Since every message sent must be received by someone, the total messages sent should equal the total messages received. So, I can set up the equation:7n + 15m = 15n + 9mNow, let's solve for n and m. I'll subtract 7n and 9m from both sides to get:15m - 9m = 15n - 7nSimplifying both sides:6m = 8nI can divide both sides by 2 to simplify further:3m = 4nSo, 3m equals 4n. To find the relationship between m and n, I can express m in terms of n:m = (4/3)nThis means that m is 4/3 times n. Since 4/3 is greater than 1, m is greater than n. Therefore, there are more employees in the iOS department than in the Android department.Wait, let me double-check my steps to make sure I didn't make a mistake. I set up the total messages sent and received correctly, right? Yes, because each message sent must be received by someone, so the totals should balance out.Then, I subtracted 7n and 9m from both sides, which seems correct. Simplifying 6m = 8n, and then dividing by 2 gives 3m = 4n. Yes, that looks right.Expressing m in terms of n gives m = (4/3)n, which means m is larger than n. So, the iOS department has more employees.I think that makes sense. The iOS developers are sending more messages per person, but receiving fewer, while Android developers are sending fewer but receiving more. So, to balance out the total messages, there must be more iOS developers to account for the higher number of messages being sent.Yeah, I think I got it right. The iOS department has more employees.

💡First, I need to determine the total number of students across all three classes by adding the number of students in each class.Next, I will calculate the total sum of marks for each class by multiplying the average mark by the number of students in that class.After obtaining the total marks for each class, I will sum these amounts to find the overall total marks for all students combined.Finally, I will divide the total combined marks by the total number of students to find the combined average mark.

💡Okay, so I've got this problem here where two objects start moving towards each other from points A and B. The total distance between A and B is m, and the product of the distances each object travels before they meet is n squared. I need to find the polynomial whose roots are these distances. The options are given as A, B, C, D, each with different quadratic expressions.Alright, let me break this down. First, let's visualize the scenario. There's point A and point B, m units apart. Object 1 starts at A, moving towards B, and Object 2 starts at B, moving towards A. They meet somewhere in between after some time. The distance Object 1 has traveled from A is, let's say, x, and the distance Object 2 has traveled from B is y. Since they meet, the sum of these distances should be equal to the total distance between A and B, which is m. So, x + y = m. That's one equation.The problem also states that the product of these distances is n squared. So, x * y = n². That's the second equation.Now, I need to find a quadratic polynomial where x and y are the roots. From what I remember, if a quadratic equation has roots α and β, then the equation can be written as (x - α)(x - β) = 0. Expanding this, we get x² - (α + β)x + αβ = 0. So, the sum of the roots is α + β, and the product is αβ.In this case, the sum of the roots x + y is m, and the product x * y is n². So, substituting these into the quadratic equation, we should get x² - (x + y)x + x * y = 0, which simplifies to x² - m x + n² = 0.Let me check the options given:A. x² - m x - n²B. x² + m x + n²C. x² - m x + n²D. x² + m x - n²So, the equation I derived is x² - m x + n², which is option C.Wait, let me make sure I didn't make a mistake. The sum of the roots is m, so in the quadratic equation, the coefficient of x is negative the sum, which is -m. The constant term is the product, which is n². So, yes, that would be x² - m x + n².Just to double-check, if I consider the roots x and y, then (x - x)(x - y) would give me the equation, but actually, it's (x - x1)(x - x2) where x1 and x2 are the roots. So, expanding (x - x1)(x - x2) gives x² - (x1 + x2)x + x1x2, which is exactly what I used.Therefore, I'm confident that the correct polynomial is x² - m x + n², which is option C.**Final Answer**boxed{C}

💡Okay, so I have this problem here about a sequence {a_n}. It says that a_1 is 2, and then each subsequent term is defined by a_{n+1} = a_n + c*n, where c is a constant. Also, it mentions that the first three terms, a_1, a_2, and a_3, form a geometric sequence with a common ratio that's not equal to 1. I need to find the value of c.Alright, let's break this down step by step. First, let's write out what we know.We know that a_1 is 2. Then, a_2 is a_1 + c*1, right? Because when n=1, a_{2} = a_1 + c*1. So, a_2 = 2 + c.Similarly, a_3 would be a_2 + c*2, because when n=2, a_{3} = a_2 + c*2. So, substituting a_2, that's (2 + c) + 2c, which simplifies to 2 + 3c.So, now we have the first three terms:a_1 = 2a_2 = 2 + ca_3 = 2 + 3cNow, it says that these three terms form a geometric sequence. In a geometric sequence, each term is the previous term multiplied by a common ratio, r. So, the ratio between a_2 and a_1 should be the same as the ratio between a_3 and a_2.Mathematically, that means:a_2 / a_1 = a_3 / a_2Let me write that out with the expressions we have:(2 + c) / 2 = (2 + 3c) / (2 + c)So, cross-multiplying to solve for c:(2 + c)^2 = 2*(2 + 3c)Let me compute both sides.Left side: (2 + c)^2 = 4 + 4c + c^2Right side: 2*(2 + 3c) = 4 + 6cSo, setting them equal:4 + 4c + c^2 = 4 + 6cSubtract 4 from both sides:4c + c^2 = 6cBring all terms to one side:c^2 + 4c - 6c = 0Simplify:c^2 - 2c = 0Factor:c(c - 2) = 0So, the solutions are c = 0 or c = 2.But wait, the problem says that the common ratio is not equal to 1. Let me check what happens if c is 0 or 2.If c = 0, then the sequence becomes:a_1 = 2a_2 = 2 + 0 = 2a_3 = 2 + 0 = 2So, all terms are 2, which is a geometric sequence with a common ratio of 1. But the problem specifies that the common ratio is not equal to 1. Therefore, c = 0 is not acceptable.If c = 2, then:a_1 = 2a_2 = 2 + 2 = 4a_3 = 4 + 4 = 8So, the sequence is 2, 4, 8, which is a geometric sequence with a common ratio of 2. That's acceptable because the ratio is not 1.Therefore, the value of c is 2.Wait, let me just double-check my calculations to make sure I didn't make a mistake.Starting with a_1 = 2.a_2 = a_1 + c*1 = 2 + ca_3 = a_2 + c*2 = (2 + c) + 2c = 2 + 3cSet up the geometric sequence condition:(2 + c)/2 = (2 + 3c)/(2 + c)Cross-multiplied:(2 + c)^2 = 2*(2 + 3c)Expanding left side: 4 + 4c + c^2Right side: 4 + 6cSubtract right side from left side:4 + 4c + c^2 - 4 - 6c = c^2 - 2c = 0Factor: c(c - 2) = 0So, c = 0 or c = 2.As before, c = 0 gives a common ratio of 1, which is invalid. So, c must be 2.I think that's correct. So, the value of c is 2.**Final Answer**The value of ( c ) is boxed{2}.