answer:To solve this problem, we need to follow these steps: Step 1: Find the Critical PointsA critical point of a function occurs where its derivative is zero or undefined. Since ( V(t) ) is a polynomial, its derivative will be defined everywhere, so we only need to find where the derivative is zero.1. Find the derivative of ( V(t) ): [ V(t) = -2t^3 + 3t^2 + 12t + 5 ] The derivative ( V'(t) ) is: [ V'(t) = frac{d}{dt}(-2t^3 + 3t^2 + 12t + 5) = -6t^2 + 6t + 12 ]2. Set the derivative equal to zero and solve for ( t ): [ -6t^2 + 6t + 12 = 0 ] Divide through by -6 to simplify: [ t^2 - t - 2 = 0 ] Factor the quadratic: [ (t - 2)(t + 1) = 0 ] Set each factor to zero: [ t - 2 = 0 quad Rightarrow quad t = 2 ] [ t + 1 = 0 quad Rightarrow quad t = -1 ] Since ( t ) represents time in months since the regulation was implemented, we only consider ( t = 2 ) as a valid critical point. Step 2: Determine the Time Intervals where ( V(t) < 15 )1. Set up the inequality: [ V(t) < 15 ] Substitute the polynomial: [ -2t^3 + 3t^2 + 12t + 5 < 15 ] Simplify: [ -2t^3 + 3t^2 + 12t + 5 - 15 < 0 ] [ -2t^3 + 3t^2 + 12t - 10 < 0 ]2. Solve the inequality: This involves finding the roots of the polynomial to determine the intervals. Solving a cubic algebraically can be complex, so let's approximate roots using numerical methods such as the Rational Root Theorem, synthetic division, or graphing calculators. We find the roots approximately to be ( t approx 0.5, 2.5, 4 ).3. Test the intervals: We test the intervals determined by these roots: ( (0, 0.5) ), ( (0.5, 2.5) ), ( (2.5, 4) ), and ( (4, 12) ). - For ( t = 1 ) (in ( (0.5, 2.5) )): ( V(1) = -2(1)^3 + 3(1)^2 + 12(1) + 5 = 18 ) (not less than 15) - For ( t = 3 ) (in ( (2.5, 4) )): ( V(3) = -2(3)^3 + 3(3)^2 + 12(3) + 5 = 5 ) (less than 15) - Test other points similarly to determine intervals.By evaluating the sign changes around these intervals, we determine that:- ( V(t) < 15 ) for ( t in (2.5, 4) ). Final AnswerThe critical point is ( t = 2 ). The time intervals within ( 0 leq t leq 12 ) where ( V(t) < 15 ) are ( t in (2.5, 4) ).Final Answer: The final answer is ( t = 2 ) and ( t in (2.5, 4) ) where ( V(t) < 15 ). I hope it is correct.

answer:To solve the problem, we need to address each part of the question step by step.1. Determine the number of followers the author had at the start of the campaign. The number of followers at the start of the campaign is given by evaluating the function ( F(t) ) at ( t = 0 ). [ F(0) = -2(0)^2 + 40(0) + 500 = 500 ] So, the author had 500 followers at the start of the campaign.2. After how many months will the author's followers reach their maximum? What is the maximum number of followers they will achieve? The function ( F(t) = -2t^2 + 40t + 500 ) is a quadratic function in the standard form ( at^2 + bt + c ), where ( a = -2 ), ( b = 40 ), and ( c = 500 ). For a quadratic equation of the form ( ax^2 + bx + c ), the vertex occurs at ( t = -frac{b}{2a} ). [ t = -frac{40}{2(-2)} = frac{40}{4} = 10 ] To find the maximum number of followers, substitute ( t = 10 ) back into the function ( F(t) ): [ F(10) = -2(10)^2 + 40(10) + 500 = -2(100) + 400 + 500 = -200 + 400 + 500 = 700 ] Therefore, the maximum number of followers is 700, and this occurs after 10 months.3. Combine the models for ( F(t) ) and ( I(x) ) to express a new function that models the total number of followers after ( t ) months, assuming ( t = x ). Simplify the expression. The total number of followers after ( t ) months, including the increase from articles, can be modeled by the sum of the two functions ( F(t) ) and ( I(x) ), where ( t = x ). [ F(t) + I(x) = (-2t^2 + 40t + 500) + (300 + 25t) ] Simplify the expression: [ = -2t^2 + 40t + 500 + 300 + 25t ] Combine like terms: [ = -2t^2 + (40t + 25t) + (500 + 300) ] [ = -2t^2 + 65t + 800 ] Therefore, the new function modeling the total number of followers is ( T(t) = -2t^2 + 65t + 800 ).Final Answer: The final answer is ( T(t) = -2t^2 + 65t + 800 ). I hope it is correct.

answer:To solve the given problem, we'll address each part separately. Part 1: Finding the roots of the polynomialWe have the polynomial ( p(x) = 2x^3 - 15x^2 + 24x - 9 ). We want to find the values of ( x ) for which ( p(x) = 0 ).1. Rational Root Theorem: According to the Rational Root Theorem, any rational root, in the form of ( frac{p}{q} ), of the polynomial is such that ( p ) divides the constant term (-9), and ( q ) divides the leading coefficient (2). Thus, the possible rational roots are ( pm 1, pm 3, pm 9, pm frac{1}{2}, pm frac{3}{2}, pm frac{9}{2} ).2. Testing Possible Roots: - Testing ( x = 1 ): [ p(1) = 2(1)^3 - 15(1)^2 + 24(1) - 9 = 2 - 15 + 24 - 9 = 2 ] Not a root. - Testing ( x = 3 ): [ p(3) = 2(3)^3 - 15(3)^2 + 24(3) - 9 = 54 - 135 + 72 - 9 = -18 ] Not a root. - Testing ( x = 2 ): [ p(2) = 2(2)^3 - 15(2)^2 + 24(2) - 9 = 16 - 60 + 48 - 9 = -5 ] Not a root. - Testing ( x = frac{3}{2} ): [ pleft(frac{3}{2}right) = 2left(frac{3}{2}right)^3 - 15left(frac{3}{2}right)^2 + 24left(frac{3}{2}right) - 9 ] [ = 2left(frac{27}{8}right) - 15left(frac{9}{4}right) + 36 - 9 ] [ = frac{54}{8} - frac{135}{4} + 36 - 9 ] [ = frac{54}{8} - frac{270}{8} + frac{216}{8} - frac{72}{8} ] [ = frac{54 + 216 - 270 - 72}{8} = 0 ] Thus, ( x = frac{3}{2} ) is a root.3. Polynomial Division: We divide ( p(x) ) by ( x - frac{3}{2} ) using synthetic or long division to find the other roots.After performing polynomial division, we obtain:[p(x) = (x - frac{3}{2})(2x^2 - 12x + 6)]4. Solving ( 2x^2 - 12x + 6 = 0 ): - Use the quadratic formula: ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a = 2, b = -12, c = 6 ). - Discriminant: ( b^2 - 4ac = (-12)^2 - 4 times 2 times 6 = 144 - 48 = 96 ). - Roots: [ x = frac{12 pm sqrt{96}}{4} ] [ x = frac{12 pm 4sqrt{6}}{4} ] [ x = 3 pm sqrt{6} ]Thus, the roots of the polynomial ( p(x) = 0 ) are ( x = frac{3}{2}, 3 + sqrt{6}, 3 - sqrt{6} ). Part 2: Solving the inequalityThe inequality to solve is ( 3x^2 - 12x + 9 < 0 ).1. Quadratic Equation: Rewrite the quadratic as ( 3(x^2 - 4x + 3) < 0 ).2. Finding the Roots: - Solve ( x

answer:To solve the given math problem, we need to address each part of the problem step by step. Part 1: Solve ( P(x) = 0 )Given the polynomial function:[ P(x) = 4x^3 - 12x^2 + 9x - 15 ]We need to find the values of ( x ) where ( P(x) = 0 ), i.e.,[ 4x^3 - 12x^2 + 9x - 15 = 0 ]To solve this cubic equation, we can attempt to find rational roots using the Rational Root Theorem, which states that any rational solution, in the form of ( frac{p}{q} ), is a factor of the constant term divided by a factor of the leading coefficient. For this polynomial, possible rational roots are the factors of (-15) divided by the factors of (4), i.e., ( pm 1, pm 3, pm 5, pm 15, pm frac{1}{2}, pm frac{3}{2}, pm frac{5}{2}, pm frac{15}{2}, pm frac{1}{4}, pm frac{3}{4}, pm frac{5}{4}, pm frac{15}{4} ).Testing these candidates, let's try ( x = 1 ):[ P(1) = 4(1)^3 - 12(1)^2 + 9(1) - 15 = 4 - 12 + 9 - 15 = -14 ]Since ( P(1) neq 0 ), ( x = 1 ) is not a root. Similarly, by testing other values, we find:Testing ( x = 3 ):[ P(3) = 4(3)^3 - 12(3)^2 + 9(3) - 15 = 108 - 108 + 27 - 15 = 12 ]It turns out ( x = 3 ) is not a root either.Continuing this process can be tedious, so let's try ( x = frac{3}{2} ):[ Pleft(frac{3}{2}right) = 4left(frac{3}{2}right)^3 - 12left(frac{3}{2}right)^2 + 9left(frac{3}{2}right) - 15 ][ = 4 cdot frac{27}{8} - 12 cdot frac{9}{4} + frac{27}{2} - 15 ][ = frac{108}{8} - frac{108}{4} + frac{54}{4} - 15 ][ = frac{27}{2} - 27 + frac{27}{2} - 15 ][ = 0 ]Thus, ( x = frac{3}{2} ) is a root. Now we divide the polynomial by ( x - frac{3}{2} ) to find the remaining roots.Perform polynomial division of ( 4x^3 - 12x^2 + 9x - 15 ) by ( 2x - 3 ):Using synthetic division or polynomial long division, we find that:[ P(x) = (2x - 3)(2x^2 - 3) ]Setting each factor to zero:1. ( 2x - 3 = 0 ) gives ( x = frac{3}{2} ).2. ( 2x^2 - 3 = 0 ) gives ( x^2 = frac{3}{2} ), so ( x = pm sqrt{frac{3}{2}} ).Thus, the solutions for ( P(x) = 0 ) are:[ x = frac{3}{2}, quad x = sqrt{frac{3}{2}}, quad x = -sqrt{frac{3}{2}} ] Part 2: Solve ( P(x) geq 10 )We need to determine the range of ( x ) for which:[ 4x^3 - 12x^2 + 9x - 15 geq 10 ]This simplifies to:[ 4x^3 - 12x^2 + 9x - 25 geq 0 ]Using the factorization from Part 1, the polynomial becomes:[ (2x - 3)(2x^2 - 3) geq 0 ]Analyze the sign of the product ( (2x - 3)(

answer:To solve the given problem, let's break it down into two parts and solve each part step-by-step. Part 1: Determine the number of classic and modern filmsWe are given the following conditions:- A total of 10 films are screened.- The total cost for screening these films is 500.- The cost for a classic film is 30, and for a modern film, it is 50.Let ( x ) be the number of classic films, and ( y ) be the number of modern films. We can set up the following system of equations based on the problem statement:1. ( x + y = 10 ) (Total number of films)2. ( 30x + 50y = 500 ) (Total cost for the films)Now, solve this system of equations.Step 1: Solve the first equation for one variable. Let's solve for ( x ):[x = 10 - y]Step 2: Substitute ( x = 10 - y ) into the second equation:[30(10 - y) + 50y = 500]Step 3: Simplify and solve for ( y ):[300 - 30y + 50y = 500][300 + 20y = 500][20y = 200][y = 10]Step 4: Substitute ( y = 10 ) back into the equation for ( x ):[x = 10 - y = 10 - 10 = 0]Thus, the cinephile can screen 0 classic films and 10 modern films. Part 2: Determine the number of attendees for each type of filmWe are given:- The ratio of audience members attending modern films to classic films is 3:1.- A total of 400 attendees watched the films.Let ( a ) be the number of attendees for modern films, and ( b ) be the number of attendees for classic films. We have the following conditions:1. ( a + b = 400 ) (Total number of attendees)2. ( frac{a}{b} = 3 ) (Ratio of modern to classic film attendees)Step 1: From the ratio, express ( a ) in terms of ( b ):[a = 3b]Step 2: Substitute ( a = 3b ) into the total number of attendees equation:[3b + b = 400][4b = 400][b = 100]Step 3: Substitute ( b = 100 ) back into the equation for ( a ):[a = 3b = 3 times 100 = 300]Thus, 300 attendees watched modern films, and 100 attendees watched classic films.Final Answer: The cinephile screens 0 classic films and 10 modern films. Additionally, 300 attendees watched modern films and 100 attendees watched classic films. I hope it is correct.

answer:To solve the given problem, we need to address both parts systematically. Part 1: Maximizing the Average ImprovementThe function given is a quadratic function:[ I(x) = -2x^2 + 12x + 5 ]This is a downward-opening parabola (since the coefficient of ( x^2 ) is negative), and its maximum value occurs at the vertex. The formula to find the ( x )-coordinate of the vertex of a parabola given by ( ax^2 + bx + c ) is:[ x = -frac{b}{2a} ]For our function:- ( a = -2 )- ( b = 12 )Plug these values into the formula:[ x = -frac{12}{2(-2)} = -frac{12}{-4} = 3 ]So, the number of resources ( x ) that maximizes the average improvement is ( x = 3 ).To find the maximum average improvement, substitute ( x = 3 ) back into the function:[ I(3) = -2(3)^2 + 12(3) + 5 ][ = -2(9) + 36 + 5 ][ = -18 + 36 + 5 ][ = 23 ]Thus, the maximum average improvement is 23 points. Part 2: Solving the Inequality ( I(x) geq 20 )We need to solve:[ -2x^2 + 12x + 5 geq 20 ]Subtract 20 from both sides to set the inequality to zero:[ -2x^2 + 12x + 5 - 20 geq 0 ][ -2x^2 + 12x - 15 geq 0 ]Let's solve the corresponding equation:[ -2x^2 + 12x - 15 = 0 ]Using the quadratic formula ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a = -2 ), ( b = 12 ), and ( c = -15 ):First, calculate the discriminant:[ b^2 - 4ac = 12^2 - 4(-2)(-15) ][ = 144 - 120 ][ = 24 ]Now, apply the quadratic formula:[ x = frac{-12 pm sqrt{24}}{2(-2)} ][ = frac{-12 pm 2sqrt{6}}{-4} ][ = frac{12 mp 2sqrt{6}}{4} ][ = 3 mp frac{sqrt{6}}{2} ]The roots are:[ x_1 = 3 + frac{sqrt{6}}{2} ][ x_2 = 3 - frac{sqrt{6}}{2} ]The inequality ( -2x^2 + 12x - 15 geq 0 ) indicates that the function is non-negative between the roots ( x_2 ) and ( x_1 ).Thus, the range of ( x ) for which ( I(x) geq 20 ) is:[ x in left[ 3 - frac{sqrt{6}}{2}, 3 + frac{sqrt{6}}{2} right] ]Final Answer: The number of resources that maximizes average improvement is 3, with a maximum improvement of 23 points. To achieve at least an average improvement of 20 points, provide between ( 3 - frac{sqrt{6}}{2} ) and ( 3 + frac{sqrt{6}}{2} ) resources. I hope it is correct.