answer:We are given the equation: [ k^2 - 2016 = 3^n ] We need to determine all positive integers ( k ) and ( n ) that satisfy this equation. Step 1: Consider the equation modulo 3 First, let's consider the equation modulo 3. Notice that ( 2016 ) is divisible by 3: [ 2016 equiv 0 pmod{3} ] Since ( 3^n ) is obviously divisible by 3 for any positive integer ( n ), we write: [ k^2 equiv 2016 + 3^n pmod{3} ] Which simplifies to: [ k^2 equiv 0 pmod{3} ] Step 2: Implication of ( k equiv 0 pmod{3} ) This tells us that ( k ) must also be divisible by 3. Let ( k = 3m ) for some integer ( m ). Substitute ( k = 3m ) into the original equation: [ (3m)^2 - 2016 = 3^n ] [ 9m^2 - 2016 = 3^n ] Step 3: Divide the entire equation by 9 By dividing through by 9, we get: [ m^2 - frac{2016}{9} = frac{3^n}{9} ] Step 4: Simplification and analysis of divisibility Calculate ( frac{2016}{9} ): [ 2016 div 9 = 224 ] Thus, we can rewrite the equation as: [ m^2 - 224 = 3^{n-2} ] Step 5: Consider modulo 3 properties of ((m / 3)^2) and (224) Note that: [ 224 equiv 2 pmod{3} ] And: [ (m / 3)^2 equiv m^2 equiv 0, 1 pmod{3} ] Thus: [ 3^{n-2} equiv m^2 - 224 pmod{3} ] [ 3^{n-2} equiv 0, 1 - 2 pmod{3} ] So, ( 3^{n-2} pmod{3} equiv 1 pmod{3} ) Step 6: Analyze ( 3^{n-2} ) However, since ( 3^{n-2} ) must equal 0 modulo 3 unless ( n-2 = 0 ): So ( n-2 = 0 implies n = 2 ) Step 7: Substitute ( n ) back into the equation If ( n = 2 ): [ m^2 - 224 = 3^{2-2} ] [ m^2 - 224 = 1 ] This yields: [ m^2 = 225 ] [ m = 15 ] (since ( m ) must be positive) Since ( k = 3m ): [ k = 3 times 15 = 45 ] # Conclusion: The solution to the problem is ( k = 45 ) and ( n = 2 ): [ boxed{(k, n) = (45, 2)} ]

answer:To find the absolute value of -5, we follow the definition of absolute value, which is the distance of a number from 0 on the number line, without considering direction. Thus, we have: [ begin{align*} text{Absolute value of } -5 &= |-5| &= 5 end{align*} ] Therefore, the correct choice is boxed{B}.

answer:If the second tank is only 45% filled with 450 liters, we can calculate the total capacity of the tank by dividing the amount of water it currently has by the percentage it represents: Total capacity of the second tank = 450 liters / 0.45 = 1000 liters Since the two tanks have the same capacity, the first tank also has a total capacity of 1000 liters. To find out how many more liters are needed to fill the first tank, we subtract the amount it already has from its total capacity: Liters needed for the first tank = Total capacity - Amount filled Liters needed for the first tank = 1000 liters - 300 liters = 700 liters The second tank is already 45% filled, so it needs an additional 55% to be full. We can calculate the liters needed to fill the second tank by finding 55% of its total capacity: Liters needed for the second tank = 55% of total capacity Liters needed for the second tank = 0.55 * 1000 liters = 550 liters Now, to find the total liters needed to fill both tanks, we add the liters needed for each tank: Total liters needed = Liters needed for the first tank + Liters needed for the second tank Total liters needed = 700 liters + 550 liters = 1250 liters Therefore, boxed{1250} more liters of water are needed to fill the two tanks.

answer:First, to solve for the maximum value of f(x), we need to express the function in a simplified form. Given that f(x) = cos 2x + 6cosleft(frac{pi}{2} - xright), let's use trigonometric identities to re-write the function. Using the double-angle formula for cosine and the co-function identity for cosleft(frac{pi}{2} - xright), we get: begin{align*} f(x) &= cos 2x + 6cosleft(frac{pi}{2} - xright) &= cos^2 x - sin^2 x + 6sin x &= (1 - sin^2 x) - sin^2 x + 6sin x &= 1 - 2sin^2 x + 6sin x. end{align*} To facilitate finding the maximum value, let's make a substitution where t = sin x. Since sin x varies between -1 and 1, t will also belong to the interval [-1, 1]. Substituting t = sin x into the expression, we have the following quadratic function in terms of t: y = -2t^2 + 6t + 1. To find the maximum value of this quadratic function, we can complete the square or find the vertex of the parabola that this equation represents. Completing the square, we have: begin{align*} y &= -2(t^2 - 3t) + 1 &= -2left(t^2 - 3t + left(frac{3}{2}right)^2right) + 2left(frac{3}{2}right)^2 + 1 &= -2left(t - frac{3}{2}right)^2 + frac{9}{2} + 1 &= -2left(t - frac{3}{2}right)^2 + frac{11}{2}. end{align*} Since the coefficient of the squared term is negative, this function opens downwards, and thus reaches its maximum at the vertex of the parabola, when t = frac{3}{2}. However, we must remember that t = sin x, and sin x cannot exceed 1. Thus, the maximum value of t is 1, not frac{3}{2}. Plugging this value back into our quadratic function: y_{text{max}} = -2(1 - frac{3}{2})^2 + frac{11}{2} = -2left(-frac{1}{2}right)^2 + frac{11}{2} = -2left(frac{1}{4}right) + frac{11}{2} = -frac{1}{2} + frac{11}{2} = frac{10}{2} = 5. So, the maximum value of the function f(x) is boxed{5}.