💡Okay, so I have this problem here about a sine function, and I need to find the absolute value of the phase shift φ after some transformation. Let me try to break it down step by step.The original function is y = 3 sin(2x + φ), and φ is between -π and 0. Then, the graph is shifted to the left by π/6, and after this shift, the resulting graph is symmetric about the y-axis. I need to find |φ|.First, I remember that shifting a graph to the left by π/6 means replacing x with x + π/6 in the function. So, let me write that out:Original function: y = 3 sin(2x + φ)After shifting left by π/6: y = 3 sin[2(x + π/6) + φ]Let me simplify that expression inside the sine:2(x + π/6) = 2x + 2*(π/6) = 2x + π/3So, the shifted function becomes: y = 3 sin(2x + π/3 + φ)Now, the problem states that this shifted graph is symmetric about the y-axis. Symmetry about the y-axis means that the function is even, right? So, for a function to be even, it must satisfy f(x) = f(-x) for all x.Let me write that condition for our shifted function:3 sin(2x + π/3 + φ) = 3 sin(-2x + π/3 + φ)Since the 3 is just a coefficient, it cancels out on both sides, so we have:sin(2x + π/3 + φ) = sin(-2x + π/3 + φ)Hmm, I need to recall the identity for when two sine functions are equal. I remember that sin(A) = sin(B) implies that either A = B + 2πn or A = π - B + 2πn for some integer n.So, applying that here, we have two cases:1. 2x + π/3 + φ = -2x + π/3 + φ + 2πn2. 2x + π/3 + φ = π - (-2x + π/3 + φ) + 2πnLet me simplify both cases.Starting with case 1:2x + π/3 + φ = -2x + π/3 + φ + 2πnSubtract π/3 + φ from both sides:2x = -2x + 2πnAdd 2x to both sides:4x = 2πnDivide both sides by 2:2x = πnSo, x = (πn)/2But this has to hold for all x, right? Because the function is symmetric about the y-axis for all x. However, x = (πn)/2 is only true for specific x values, not for all x. Therefore, this case doesn't provide a valid solution because it doesn't satisfy the condition for all x. So, case 1 is not possible.Now, moving on to case 2:2x + π/3 + φ = π - (-2x + π/3 + φ) + 2πnLet me simplify the right-hand side:π - (-2x + π/3 + φ) = π + 2x - π/3 - φSo, the equation becomes:2x + π/3 + φ = π + 2x - π/3 - φ + 2πnLet me subtract 2x from both sides:π/3 + φ = π - π/3 - φ + 2πnSimplify the right-hand side:π - π/3 = (3π/3 - π/3) = 2π/3So, we have:π/3 + φ = 2π/3 - φ + 2πnNow, let's bring like terms together. I'll add φ to both sides and subtract π/3 from both sides:π/3 + φ + φ = 2π/3 + 2πnWhich simplifies to:2φ = 2π/3 - π/3 + 2πn2φ = π/3 + 2πnDivide both sides by 2:φ = π/6 + πnNow, we have φ expressed in terms of n, where n is any integer. But we have a constraint on φ: -π < φ < 0. So, let's find the value(s) of n that satisfy this condition.Let's consider different integer values for n:1. If n = 0: φ = π/6 ≈ 0.523 radians. But φ must be less than 0, so this doesn't work.2. If n = -1: φ = π/6 - π = π/6 - 6π/6 = -5π/6 ≈ -2.618 radians. This is within the range -π < φ < 0 because -π is approximately -3.1416, so -5π/6 is greater than -π.3. If n = -2: φ = π/6 - 2π = π/6 - 12π/6 = -11π/6 ≈ -5.7596 radians. This is less than -π, so it doesn't satisfy the condition.So, the only valid solution within the given range is φ = -5π/6.Therefore, the absolute value of φ is |φ| = 5π/6.Let me just double-check my steps to make sure I didn't make a mistake.1. Shifted the function left by π/6: Correct, replaced x with x + π/6.2. Simplified the expression inside sine: 2x + π/3 + φ: Correct.3. Set up the condition for even function: sin(2x + π/3 + φ) = sin(-2x + π/3 + φ): Correct.4. Applied the sine equality condition: Either A = B + 2πn or A = π - B + 2πn: Correct.5. Solved case 1: Ended up with x = (πn)/2, which isn't valid for all x: Correct.6. Solved case 2: Ended up with φ = π/6 + πn: Correct.7. Tested n values: n=0 gives positive φ, which is invalid; n=-1 gives φ=-5π/6, which is valid; n=-2 gives φ less than -π, invalid: Correct.Everything seems to check out. So, I'm confident that |φ| is 5π/6.**Final Answer**boxed{D}

💡First, I'll define variables for each type of item Janice purchased. Let ( a ) be the number of 50-cent items, ( b ) the number of 1.50 items, ( c ) the number of 2.50 items, and ( d ) the number of 3.50 items.Given that Janice bought a total of 50 items, I can write the equation:[a + b + c + d = 50]The total cost of the items is 100.00, which is 10000 cents. Converting the prices to cents, the total cost equation becomes:[50a + 150b + 250c + 350d = 10000]To simplify, I'll divide the entire cost equation by 50:[a + 3b + 5c + 7d = 200]Now, I'll substitute ( a ) from the first equation into the simplified cost equation:[50 - (b + c + d) + 3b + 5c + 7d = 200][50 + 2b + 4c + 6d = 200][2b + 4c + 6d = 150]Dividing this equation by 2 to further simplify:[b + 2c + 3d = 75]To find specific values for ( b ), ( c ), and ( d ), I'll assume ( b = 15 ), ( c = 15 ), and ( d = 15 ). These values satisfy the equation:[15 + 2(15) + 3(15) = 15 + 30 + 45 = 90]Finally, I'll calculate ( a ) using the total number of items:[a = 50 - (15 + 15 + 15) = 5]Therefore, Janice purchased 5 items costing 50 cents each.

💡Okay, so I have this problem about vectors. It says that given two vectors, a and b, with the conditions that (a + b) squared minus b squared equals the magnitude of a, which is 3, and also that the magnitude of b is at least 2. I need to find the range of the projection of a onto b.Hmm, let me start by understanding what each part means. First, the condition (a + b)^2 - b^2 = |a| = 3. I know that when you square a vector, it's the dot product of the vector with itself. So, (a + b)^2 would be (a + b) · (a + b). Similarly, b^2 is b · b.So, expanding (a + b) · (a + b), that should be a · a + 2a · b + b · b. Then, subtracting b · b from that, we get a · a + 2a · b. And this equals |a|, which is 3. Wait, |a| is the magnitude of a, so |a| = 3. But a · a is |a|^2, right? So, |a|^2 is 9.So, substituting back, 9 + 2a · b = 3. That means 2a · b = 3 - 9, which is -6. So, a · b = -3. Okay, got that.Now, the projection of a onto b is given by the formula |a| cos(theta), where theta is the angle between a and b. Alternatively, it can be written as (a · b)/|b|. Since we have a · b = -3, the projection is -3 divided by |b|.But we also know that |b| is at least 2. So, |b| >= 2. Therefore, the projection is -3 / |b|, and since |b| is at least 2, the projection will be between -3/2 and 0. Because as |b| increases, -3 / |b| approaches 0 from the negative side. So, when |b| is exactly 2, the projection is -3/2, and as |b| gets larger, the projection becomes less negative, approaching 0 but never reaching it.Wait, let me make sure I'm not making a mistake here. The projection is a scalar, right? So, it can be positive or negative depending on the angle between the vectors. Since a · b is negative, that means the angle between a and b is obtuse, so the projection is negative. That makes sense.So, the projection is -3 / |b|, and since |b| >= 2, the smallest |b| can be is 2, giving the projection as -3/2. As |b| increases beyond 2, the projection becomes less negative, approaching 0. So, the range of the projection should be from -3/2 up to, but not including, 0.Is there a possibility that |b| could be exactly 2? The problem says |b| >= 2, so yes, |b| can be 2. Therefore, -3/2 is included in the range. But since |b| can be any number greater than or equal to 2, the projection can get as close to 0 as we want, but it will never actually reach 0 because that would require |b| to be infinite, which isn't the case here.So, putting it all together, the projection of a onto b is between -3/2 and 0, including -3/2 but not including 0. Therefore, the range is [-3/2, 0).I think that makes sense. Let me just recap to make sure I didn't skip any steps or make any errors. Starting from the given condition, expanding the vectors, solving for the dot product, then using the projection formula, considering the constraints on |b|, and determining the range accordingly. Yeah, that seems solid.I don't think I made any mistakes in the calculations. The key steps were expanding the square of the vector sum, recognizing that a · a is |a|^2, solving for the dot product, and then using that to find the projection. Then, applying the constraint on |b| to find the bounds of the projection.So, I feel confident that the range of the projection is from -3/2 to 0, not including 0.

💡First, I'll denote the usual time it takes for the man to reach his office as ( T ) (in hours) and his usual rate (speed) as ( R ) (in distance per hour).When the man walks at ( frac{3}{4} ) of his usual rate, his new rate becomes ( frac{3}{4}R ).Since the time it takes to travel a certain distance is inversely proportional to the rate of travel, walking at a slower rate will take him longer. Let's call this longer time ( T' ).We know that ( T' = T + frac{1}{3} ) (since he is ( frac{1}{3} ) of an hour late).The relationship between time, rate, and distance is given by the formula:[text{Distance} = text{Rate} times text{Time}]The distance to the office is the same in both cases, so we can set up the equation:[R times T = frac{3}{4}R times T']Substituting ( T' ) with ( T + frac{1}{3} ):[T = frac{3}{4}(T + frac{1}{3})]Distributing the ( frac{3}{4} ):[T = frac{3}{4}T + frac{3}{4} times frac{1}{3}][T = frac{3}{4}T + frac{1}{4}]Subtracting ( frac{3}{4}T ) from both sides:[T - frac{3}{4}T = frac{1}{4}][frac{1}{4}T = frac{1}{4}]Dividing both sides by ( frac{1}{4} ):[T = 1]

💡First, I need to determine the number of goats in the caravan. I know there are 50 hens, 8 camels, and 15 keepers. The total number of feet is 224 more than the number of heads.I'll start by calculating the total number of heads. This includes the heads of hens, goats, camels, and keepers:[text{Total heads} = 50 text{ (hens)} + G text{ (goats)} + 8 text{ (camels)} + 15 text{ (keepers)} = 73 + G]Next, I'll calculate the total number of feet. Hens have 2 feet, goats have 4 feet, camels have 4 feet, and keepers have 2 feet:[text{Total feet} = 2 times 50 + 4 times G + 4 times 8 + 2 times 15 = 100 + 4G + 32 + 30 = 162 + 4G]According to the problem, the total number of feet is 224 more than the total number of heads:[162 + 4G = (73 + G) + 224]Simplifying the equation:[162 + 4G = 297 + G]Subtracting ( G ) from both sides:[162 + 3G = 297]Subtracting 162 from both sides:[3G = 135]Finally, dividing both sides by 3:[G = 45]Therefore, there are 45 goats in the caravan.

💡Alright, so I have this problem about quadratic polynomials. It says that after adding the coefficients ( p ) and ( q ) with the roots of the quadratic polynomial ( x^2 + px + q ), the result is 2, and after multiplying them, the result is 12. I need to find all such quadratic polynomials.Hmm, okay. Let me try to break this down step by step. First, I know that a quadratic polynomial generally looks like ( x^2 + px + q ), where ( p ) and ( q ) are coefficients. The roots of this polynomial can be found using the quadratic formula, but maybe I can use Vieta's formulas here because they relate the coefficients to the sum and product of the roots.Vieta's formulas say that if the roots are ( a ) and ( b ), then:- The sum of the roots ( a + b = -p )- The product of the roots ( ab = q )Okay, so that's useful. Now, the problem mentions adding the coefficients ( p ) and ( q ) with the roots. I think this means adding ( p ), ( q ), ( a ), and ( b ) together. So, the sum ( a + b + p + q = 2 ). And when we multiply them, it's ( abpq = 12 ).Wait, let me make sure I understand. The first condition is that the sum of the coefficients ( p ) and ( q ) with the roots ( a ) and ( b ) is 2. So, that would be ( a + b + p + q = 2 ). The second condition is that the product of the coefficients ( p ) and ( q ) with the roots ( a ) and ( b ) is 12, so ( abpq = 12 ).Got it. So, I have two equations:1. ( a + b + p + q = 2 )2. ( abpq = 12 )But from Vieta's formulas, I already know that ( a + b = -p ) and ( ab = q ). Maybe I can substitute these into the equations to eliminate ( a ) and ( b ).Let's substitute ( a + b ) with ( -p ) in the first equation:( -p + p + q = 2 )Simplify that: ( 0 + q = 2 ), so ( q = 2 ).Okay, so ( q = 2 ). Now, let's substitute ( ab ) with ( q ) in the second equation:( q cdot p cdot q = 12 )Wait, that would be ( q cdot p cdot q = p q^2 = 12 )Since ( q = 2 ), substitute that in:( p cdot 2^2 = 12 )Which simplifies to ( p cdot 4 = 12 )So, ( p = 12 / 4 = 3 )Alright, so ( p = 3 ) and ( q = 2 ). Therefore, the quadratic polynomial is ( x^2 + 3x + 2 ).Let me double-check to make sure this satisfies the original conditions. First, find the roots of ( x^2 + 3x + 2 ). Factoring, we get ( (x + 1)(x + 2) ), so the roots are ( x = -1 ) and ( x = -2 ).Now, let's verify the sum condition:( a + b + p + q = (-1) + (-2) + 3 + 2 = (-3) + 5 = 2 ). That checks out.Next, the product condition:( abpq = (-1)(-2)(3)(2) = (2)(6) = 12 ). That also checks out.So, it seems like ( x^2 + 3x + 2 ) is the correct polynomial. But the problem says "find all such quadratic polynomials." Is there more than one?Wait, let me think. I assumed that the polynomial is monic, meaning the coefficient of ( x^2 ) is 1. If the polynomial isn't monic, it would be ( ax^2 + bx + c ), but in the problem statement, it's given as ( x^2 + px + q ), so I think it's safe to assume it's monic. Therefore, there's only one such polynomial.But just to be thorough, let me consider if there could be another polynomial. Suppose the polynomial isn't monic, but the problem specifies ( x^2 + px + q ), so the leading coefficient is 1. Therefore, there's only one solution.So, I think I've found the only quadratic polynomial that satisfies the given conditions.**Final Answer**The quadratic polynomial is boxed{x^2 + 3x + 2}.