💡First, I identify the weights of each plastic ring:- Orange ring: 0.08333333333333333 ounce- Purple ring: 0.3333333333333333 ounce- White ring: 0.4166666666666667 ounceNext, I add these weights together to find the total weight:0.08333333333333333 + 0.3333333333333333 + 0.4166666666666667Finally, the sum of these values is 0.8333333333333333 ounce.

💡Okay, so I have this system of equations:1. (x^2 + x^2y^2 + x^2y^4 = 525)2. (x + xy + xy^2 = 35)I need to find the sum of the real (y) values that satisfy both equations. The answer choices are given, so that might help if I get stuck.First, looking at the second equation: (x + xy + xy^2 = 35). I notice that each term has an (x), so maybe I can factor that out. Let me try that:(x(1 + y + y^2) = 35)Okay, so that simplifies to (x = frac{35}{1 + y + y^2}). That seems useful because I can substitute this expression for (x) into the first equation. Let me write that down:First equation becomes:[left(frac{35}{1 + y + y^2}right)^2 + left(frac{35}{1 + y + y^2}right)^2 y^2 + left(frac{35}{1 + y + y^2}right)^2 y^4 = 525]Hmm, that looks a bit complicated, but maybe I can factor out the common term. Let's see:Factor out (left(frac{35}{1 + y + y^2}right)^2):[left(frac{35}{1 + y + y^2}right)^2 (1 + y^2 + y^4) = 525]Alright, so now I have:[frac{35^2 (1 + y^2 + y^4)}{(1 + y + y^2)^2} = 525]Calculating (35^2) gives me 1225, so substituting that in:[frac{1225 (1 + y^2 + y^4)}{(1 + y + y^2)^2} = 525]To simplify this, I can divide both sides by 525:[frac{1225}{525} times frac{(1 + y^2 + y^4)}{(1 + y + y^2)^2} = 1]Calculating (frac{1225}{525}), I get (frac{49}{21}) which simplifies to (frac{7}{3}). So now the equation is:[frac{7}{3} times frac{(1 + y^2 + y^4)}{(1 + y + y^2)^2} = 1]Multiplying both sides by (frac{3}{7}) to isolate the fraction:[frac{(1 + y^2 + y^4)}{(1 + y + y^2)^2} = frac{3}{7}]Hmm, this still looks tricky. Maybe I can cross-multiply to eliminate the denominator:[7(1 + y^2 + y^4) = 3(1 + y + y^2)^2]Expanding the right side:First, expand ((1 + y + y^2)^2):[(1 + y + y^2)^2 = 1 + 2y + 3y^2 + 2y^3 + y^4]Wait, let me check that expansion again. Let me do it step by step:- First, multiply (1) by each term: (1 times 1 = 1), (1 times y = y), (1 times y^2 = y^2).- Then, multiply (y) by each term: (y times 1 = y), (y times y = y^2), (y times y^2 = y^3).- Then, multiply (y^2) by each term: (y^2 times 1 = y^2), (y^2 times y = y^3), (y^2 times y^2 = y^4).Now, adding all these up:- Constants: 1- (y) terms: (y + y = 2y)- (y^2) terms: (y^2 + y^2 + y^2 = 3y^2)- (y^3) terms: (y^3 + y^3 = 2y^3)- (y^4) term: (y^4)So, ((1 + y + y^2)^2 = 1 + 2y + 3y^2 + 2y^3 + y^4).Therefore, the right side becomes:[3(1 + 2y + 3y^2 + 2y^3 + y^4) = 3 + 6y + 9y^2 + 6y^3 + 3y^4]Now, the left side is:[7(1 + y^2 + y^4) = 7 + 7y^2 + 7y^4]So, putting it all together:[7 + 7y^2 + 7y^4 = 3 + 6y + 9y^2 + 6y^3 + 3y^4]Let me bring all terms to one side to set the equation to zero:[7 + 7y^2 + 7y^4 - 3 - 6y - 9y^2 - 6y^3 - 3y^4 = 0]Simplify term by term:- Constants: (7 - 3 = 4)- (y) terms: (-6y)- (y^2) terms: (7y^2 - 9y^2 = -2y^2)- (y^3) terms: (-6y^3)- (y^4) terms: (7y^4 - 3y^4 = 4y^4)So, the equation becomes:[4y^4 - 6y^3 - 2y^2 - 6y + 4 = 0]Hmm, this is a quartic equation. Quartic equations can be tough, but maybe it factors nicely. Let me see if I can factor this.First, let me factor out a common factor if possible. Looking at the coefficients: 4, -6, -2, -6, 4. I don't see a common factor other than 2, but let me check:- 4y^4 - 6y^3 - 2y^2 - 6y + 4If I factor out a 2, it would be:[2(2y^4 - 3y^3 - y^2 - 3y + 2) = 0]But since 2 ≠ 0, we can ignore the 2 and focus on:[2y^4 - 3y^3 - y^2 - 3y + 2 = 0]Now, let's try to factor this quartic. Maybe it factors into quadratics or has rational roots.Using the Rational Root Theorem, possible rational roots are factors of the constant term (2) over factors of the leading coefficient (2). So possible roots are ±1, ±2, ±1/2.Let me test y = 1:[2(1)^4 - 3(1)^3 - (1)^2 - 3(1) + 2 = 2 - 3 - 1 - 3 + 2 = -3 ≠ 0]y = -1:[2(-1)^4 - 3(-1)^3 - (-1)^2 - 3(-1) + 2 = 2 + 3 - 1 + 3 + 2 = 9 ≠ 0]y = 2:[2(16) - 3(8) - 4 - 6 + 2 = 32 - 24 - 4 - 6 + 2 = 0]Wait, 32 -24 is 8, 8 -4 is 4, 4 -6 is -2, -2 +2 is 0. So y=2 is a root.Great, so (y - 2) is a factor. Let's perform polynomial division or use synthetic division to factor it out.Using synthetic division for y=2:Coefficients: 2 | -3 | -1 | -3 | 2Bring down the 2.Multiply 2 by 2: 4. Add to next coefficient: -3 + 4 = 1.Multiply 1 by 2: 2. Add to next coefficient: -1 + 2 = 1.Multiply 1 by 2: 2. Add to next coefficient: -3 + 2 = -1.Multiply -1 by 2: -2. Add to last coefficient: 2 + (-2) = 0.So the result is 2y^3 + y^2 + y -1.So, we have:[(y - 2)(2y^3 + y^2 + y - 1) = 0]Now, let's try to factor the cubic: (2y^3 + y^2 + y - 1).Again, using Rational Root Theorem. Possible roots: ±1, ±1/2.Testing y=1:[2(1)^3 + (1)^2 + 1 -1 = 2 +1 +1 -1 = 3 ≠ 0]y= -1:[2(-1)^3 + (-1)^2 + (-1) -1 = -2 +1 -1 -1 = -3 ≠ 0]y=1/2:[2(1/2)^3 + (1/2)^2 + (1/2) -1 = 2(1/8) + 1/4 + 1/2 -1 = 1/4 + 1/4 + 1/2 -1 = (1/4 +1/4)=1/2 +1/2=1 -1=0]Yes, y=1/2 is a root.So, (y - 1/2) is a factor. Let's perform synthetic division on the cubic with y=1/2.Coefficients: 2 | 1 | 1 | -1Bring down the 2.Multiply 2 by 1/2: 1. Add to next coefficient:1 +1=2.Multiply 2 by 1/2:1. Add to next coefficient:1 +1=2.Multiply 2 by 1/2:1. Add to last coefficient: -1 +1=0.So, the cubic factors into (y - 1/2)(2y^2 + 2y + 2).Thus, the quartic factors as:[(y - 2)(y - 1/2)(2y^2 + 2y + 2) = 0]Now, set each factor equal to zero:1. (y - 2 = 0) ⇒ (y = 2)2. (y - 1/2 = 0) ⇒ (y = 1/2)3. (2y^2 + 2y + 2 = 0)Let's solve the quadratic equation (2y^2 + 2y + 2 = 0). Using the quadratic formula:[y = frac{-2 pm sqrt{(2)^2 - 4(2)(2)}}{2(2)} = frac{-2 pm sqrt{4 - 16}}{4} = frac{-2 pm sqrt{-12}}{4}]Since the discriminant is negative, there are no real solutions from this quadratic. So, the only real solutions are y=2 and y=1/2.Therefore, the real y values are 2 and 1/2. The sum is (2 + 1/2 = 5/2).So, the answer should be 5/2, which is option B.**Final Answer**boxed{B}

💡Okay, so I have this problem where the solution set of the inequality ( ax^2 + bx + c > 0 ) is given as ( {x | -2 < x < 4} ). I need to figure out the relationship between the values of the function ( f(x) = ax^2 + bx + c ) at specific points: 5, 2, and -1. The options are A, B, C, D, each giving a different order of these function values.First, let me recall what the solution set ( -2 < x < 4 ) tells me about the quadratic inequality. Since the inequality ( ax^2 + bx + c > 0 ) holds between -2 and 4, that means the quadratic opens downward. If it opened upward, the inequality would hold outside the roots, not between them. So, that tells me that the coefficient ( a ) must be negative.Next, the roots of the quadratic equation ( ax^2 + bx + c = 0 ) are at ( x = -2 ) and ( x = 4 ). So, the quadratic can be factored as ( a(x + 2)(x - 4) ). Let me write that out:( f(x) = a(x + 2)(x - 4) ).Expanding this, I get:( f(x) = a(x^2 - 4x + 2x - 8) = a(x^2 - 2x - 8) ).Hmm, so ( f(x) = a(x^2 - 2x - 8) ). Since ( a ) is negative, the parabola opens downward.Now, I remember that for a quadratic function ( f(x) = ax^2 + bx + c ), the vertex is at ( x = -frac{b}{2a} ). Let me calculate that for this function.From the expanded form, ( f(x) = a(x^2 - 2x - 8) ), so ( b = -2a ). Therefore, the x-coordinate of the vertex is:( x = -frac{b}{2a} = -frac{-2a}{2a} = frac{2a}{2a} = 1 ).So, the vertex is at ( x = 1 ). Since the parabola opens downward, this vertex is the maximum point of the function.Now, I need to evaluate ( f(5) ), ( f(2) ), and ( f(-1) ). Let me think about how these points relate to the vertex at ( x = 1 ).The function is symmetric around ( x = 1 ). So, points equidistant from 1 on either side will have the same function value. For example, ( f(1 + t) = f(1 - t) ) for any ( t ).Let me compute the distances of 5, 2, and -1 from the vertex at 1:- For ( x = 5 ): distance is ( |5 - 1| = 4 ).- For ( x = 2 ): distance is ( |2 - 1| = 1 ).- For ( x = -1 ): distance is ( |-1 - 1| = 2 ).So, ( x = 5 ) is 4 units away, ( x = -1 ) is 2 units away, and ( x = 2 ) is 1 unit away from the vertex.Since the parabola opens downward, the further a point is from the vertex, the smaller the function value. So, the closer a point is to the vertex, the larger the function value.Therefore, ordering these points by their distance from the vertex:- Closest: ( x = 2 ) (distance 1)- Next: ( x = -1 ) (distance 2)- Farthest: ( x = 5 ) (distance 4)Since the function decreases as we move away from the vertex, the function values will be in the reverse order of their distances. So:( f(2) > f(-1) > f(5) ).Looking back at the options:A: ( f(5) < f(2) < f(-1) ) → This is incorrect because ( f(2) ) is actually the largest.B: ( f(-1) < f(5) < f(2) ) → This is incorrect because ( f(5) ) is the smallest.C: ( f(2) < f(-1) < f(5) ) → This is incorrect because ( f(5) ) is the smallest, not the largest.D: ( f(5) < f(-1) < f(2) ) → This is correct because ( f(5) ) is the smallest, ( f(-1) ) is in the middle, and ( f(2) ) is the largest.Wait, hold on. Let me double-check. I thought since the parabola opens downward, the function values decrease as we move away from the vertex. So, the closer to the vertex, the higher the value. So:- ( x = 2 ) is closest, so ( f(2) ) is the largest.- ( x = -1 ) is next, so ( f(-1) ) is next.- ( x = 5 ) is farthest, so ( f(5) ) is the smallest.Therefore, the order is ( f(5) < f(-1) < f(2) ), which is option D.Just to make sure, let me compute the actual function values. Since ( f(x) = a(x^2 - 2x - 8) ) and ( a < 0 ), let's pick a specific value for ( a ) to test. Let me choose ( a = -1 ) for simplicity.So, ( f(x) = -1(x^2 - 2x - 8) = -x^2 + 2x + 8 ).Compute ( f(5) ):( f(5) = -(5)^2 + 2(5) + 8 = -25 + 10 + 8 = -7 ).Compute ( f(2) ):( f(2) = -(2)^2 + 2(2) + 8 = -4 + 4 + 8 = 8 ).Compute ( f(-1) ):( f(-1) = -(-1)^2 + 2(-1) + 8 = -1 - 2 + 8 = 5 ).So, ( f(5) = -7 ), ( f(-1) = 5 ), ( f(2) = 8 ).Therefore, ( f(5) < f(-1) < f(2) ), which is indeed option D.I think that's solid. So, the answer should be D.**Final Answer**boxed{D}

💡First, I need to determine the total number of students across all grades. There are 240 freshmen, 260 sophomores, and 300 juniors. Adding these together gives a total of 800 students.Next, I'll calculate the proportion of sophomores in the entire student population. The number of sophomores is 260, so the proportion is 260 divided by 800, which simplifies to 13/40.Now, I need to apply this proportion to the total number of spots available for the study trip, which is 40. Multiplying 13/40 by 40 gives the number of sophomores that should be allocated spots on the trip.Finally, the calculation shows that 13 sophomores will be participating in the study trip.

💡First, I need to determine the number of schools in District C. Since District C has an equal number of each type of school, I'll start by identifying the smallest number of schools in any category, which is 9 private independent schools. Therefore, District C has 9 public schools, 9 parochial schools, and 9 private independent schools, totaling 27 schools.Next, I'll calculate the total number of schools in Districts A and C combined. District A has 18 schools, and District C has 27 schools, so together they have 45 schools.Finally, to find the number of schools in District B, I'll subtract the total number of schools in Districts A and C from the overall total of 50 schools. This gives me 50 - 45 = 5 schools in District B.

💡First, I need to evaluate the inner absolute value expression ( |pi - 7| ). Since ( pi ) is approximately 3.14, which is less than 7, ( pi - 7 ) is negative. Therefore, ( |pi - 7| = 7 - pi ).Next, I substitute ( |pi - 7| ) back into the original expression, resulting in ( |pi - (7 - pi)| ). Simplifying inside the absolute value gives ( |pi - 7 + pi| = |2pi - 7| ).Finally, I evaluate ( |2pi - 7| ). Since ( 2pi ) is approximately 6.28, which is less than 7, ( 2pi - 7 ) is negative. Therefore, ( |2pi - 7| = 7 - 2pi ).Thus, the exact value of the expression is ( 7 - 2pi ).