answer:Let's denote the number of gallons of water Roman used as ( R ) and the number of gallons Remy used as ( M ). According to the problem:1. Remy used 1 more gallon than 3 times the number of gallons Roman used for his shower:[ M = 3R + 1 ]2. Together, they used 33 gallons of water:[ R + M = 33 ]We need to solve these equations to find the value of ( M ). Substitute the expression for ( M ) from the first equation into the second equation:[ R + (3R + 1) = 33 ]Simplify and solve for ( R ):[ 4R + 1 = 33 ][ 4R = 32 ][ R = 8 ]Now that we have ( R ), we can use it to find ( M ):[ M = 3R + 1 ][ M = 3(8) + 1 ][ M = 24 + 1 ][ M = 25 ]So, Remy used 25 gallons of water. Let's confirm this solution using Python code for accuracy.```python# Define the given equations and solve them using Python# Variable R represents the gallons of water Roman used# Variable M represents the gallons of water Remy used# Given:# M = 3*R + 1# R + M = 33from sympy import symbols, Eq, solve# Define variablesR, M = symbols('R M')# Setup the equationsequation1 = Eq(M, 3*R + 1)equation2 = Eq(R + M, 33)# Solve the equationssolution = solve((equation1, equation2), (R, M))print(solution)``````output{R: 8, M: 25}```The solution to the problem is:[ boxed{25} ]Remy used 25 gallons of water.

answer:Let's solve the problem step by step using Python and SymPy.1. Understand the Function and Constraints: The function given is ( f(x) = a sin x + b ). We know its maximum value is 1 and minimum value is -7. Therefore, we can write: [ a sin x + b = 1 ] [ a sin x + b = -7 ]2. Solve for (a) and (b): From the equations above, we can derive: [ a sin x + b = 1 ] [ a sin x + b = -7 ] Subtracting these two equations: [ 0 = 1 - (-7) ] [ 0 = 1 + 7 ] [ 0 = 8 ] This implies that: [ a sin x = sin x ] [ b = -3 ]3. Find the Maximum Value of the Given Expression: We need to find the maximum value of ( g(x) = b sin^2 x - a cos^2 x ).Given that (b = -3) and (a sin x = 1 sin x):This implies (a = 8), substituting these values in the original expression for (b) and solving for the maximum value of (g(x)):[ g(x) = b sin^2 x - a cos^2 x ]Let's implement this in Python to find the maximum value.```pythonimport sympy as sp# Define the variable and constantsx = sp.symbols('x')a = 8b = -3# Define the expressiong = b * sp.sin(x)2 - a * sp.cos(x)2# Find the critical points by differentiating and solving for 0g_diff = sp.diff(g, x)critical_points = sp.solve(g_diff, x)# Evaluate the expression at the critical points and endpointsmax_value = float("-inf")for point in critical_points: value = g.subs(x, point) if value > max_value: max_value = value# Also check the values at the endpoints, if neededendpoints = [0, sp.pi]for point in endpoints: value = g.subs(x, point) if value > max_value: max_value = valueprint(max_value)``````output-3```The output suggests that the maximum value of ( g(x) = b sin^2 x - a cos^2 x ) is (-3). However, let's reconcile this result with the given options and the constraints we have:1. The function ( f(x) = a sin x + b ) has its maximum value at 1 and minimum value at -7.2. Given these constraints, we determined ( a = 8 ) and ( b = -3 ).Let's now reconsider the problem and verify if there are any additional critical points or if we missed something in the interpretation:To evaluate the maximum value correctly, we may need to analyze the behavior of the function ( g(x) ) over the entire domain and consider special points.The expression ( g(x) ) can be rewritten using trigonometric identities:[ g(x) = b sin^2 x - a cos^2 x ]Using (sin^2 x + cos^2 x = 1), we can rewrite ( g(x) ) as:[ g(x) = b ( 1 - cos^2 x ) - a cos^2 x ][ g(x) = b - (a + b) cos^2 x ]With ( a = 8 ) and ( b = -3):[ g(x) = -3 - (8 - 3) cos^2 x ][ g(x) = -3 - 5 cos^2 x ]Since (cos^2 x) ranges from 0 to 1:[ g(x) = -3 - 5 cos^2 x ]The maximum value of ( b sin^2 x - a cos^2 x ) occurs when (cos^2 x) is minimized, which is at (cos^2 x = 0).[ g(x) = -3 - 5 cdot 0 = -3 ]The minimum value occurs when (cos^2 x) is maximized, which is at (cos^2 x = 1).[ g(x) = -3 - 5 cdot 1 = -3 - 5 = -8 ]Based on this analysis, the maximum value is indeed (-3).So the correct answer is:[ boxed{-3} ]Given the options, the correct choice among the options provided is:(D) 5 or -3

answer:To solve the given expression ((-3)^{0} + sqrt{8} + (-3)^{2} - 4 times frac{sqrt{2}}{2}), we can break it down into the following steps:1. Compute ((-3)^{0}).2. Compute (sqrt{8}).3. Compute ((-3)^{2}).4. Compute (4 times frac{sqrt{2}}{2}).5. Sum the results of steps 1, 2, and 3, and subtract the result of step 4.Let's implement these steps using Python and SymPy to ensure an accurate result.```pythonimport sympy as sp# Define the components of the expressionterm1 = (-3)0term2 = sp.sqrt(8)term3 = (-3)2term4 = 4 * (sp.sqrt(2) / 2)# Calculate the final resultresult = term1 + term2 + term3 - term4# Print the final resultprint(result)``````output10```The calculated result for the expression ((-3)^{0} + sqrt{8} + (-3)^{2} - 4 times frac{sqrt{2}}{2}) is (boxed{10}).

answer:To determine the value of ( a ) that makes the function ( f(x) = x^3 (a cdot 2^x - 2^{-x}) ) an even function, we need to use the property of even functions. A function ( f(x) ) is even if ( f(x) = f(-x) ) for all ( x ).Let's proceed step-by-step:1. Define the function and its negative: [ f(x) = x^3 (a cdot 2^x - 2^{-x}) ] [ f(-x) = (-x)^3 (a cdot 2^{-x} - 2^{x}) = -x^3 (a cdot 2^{-x} - 2^{x}) ]2. Set ( f(x) = f(-x) ): [ x^3 (a cdot 2^x - 2^{-x}) = -x^3 (a cdot 2^{-x} - 2^{x}) ]3. Simplify the equation: As long as ( x neq 0 ), we can divide both sides by ( x^3 ) (since ( x^3 neq 0 ) when ( x neq 0 )): [ a cdot 2^x - 2^{-x} = -(a cdot 2^{-x} - 2^x) ]4. Expand and collect like terms: [ a cdot 2^x - 2^{-x} = -a cdot 2^{-x} + 2^x ] [ a cdot 2^x + a cdot 2^{-x} = 2^x + 2^{-x} ]5. Factor out terms: [ a (2^x + 2^{-x}) = 2^x + 2^{-x} ]6. Isolate ( a ): Since ( 2^x + 2^{-x} neq 0 ) for any real number ( x ): [ a = 1 ]Now let's verify this using Python and SymPy.```pythonimport sympy as sp# Define the variablesx = sp.symbols('x')a = sp.symbols('a')# Define the function f(x)f_x = x3 * (a * 2x - 2(-x))# Define f(-x)f_neg_x = (-x)3 * (a * 2(-x) - 2x)# Simplify both expressionsf_x_simplified = sp.simplify(f_x)f_neg_x_simplified = sp.simplify(f_neg_x)# Solve the equation f(x) = f(-x)a_value = sp.solve(f_x_simplified - f_neg_x_simplified, a)print(a_value)``````output[1]```The value of ( a ) that makes the function ( f(x) = x^3 (a cdot 2^x - 2^{-x}) ) an even function is (boxed{1}).

answer:To solve this problem using Python and sympy, we will take the following steps:1. Import the necessary library.2. Use sympy to calculate the square root of 9.3. Print the result.Let's implement this in code.```pythonimport sympy as sp# Define the number whose square root we want to calculatenumber = 9# Calculate the square root using sympysqrt_value = sp.sqrt(number)# Print the resultprint(sqrt_value)``````output3```The value of (sqrt{9}) is (boxed{3}).

answer:To solve this problem, we need to determine the sum of the capacities of all even subsets of ( S_4 = {1, 2, 3, 4} ). Let's break it down step-by-step:1. Understand the Sets and Capacities: - The set ( S_4 ) contains the elements ({1, 2, 3, 4}). - A subset ( X ) of ( S_4 ) can have different sets of elements. We need to calculate the capacity of each subset. - The capacity of a subset is the product of its elements.2. Identify Even and Odd Subsets: - A subset ( X ) is even if its capacity (product of its elements) is even. - A subset is odd if its capacity is odd.3. Calculate Capacities: - We need to compute the capacity for each subset and check if it is even.4. Sum the Capacities of Even Subsets: - We need to sum up the capacities of all subsets that are even.Let's write the code to perform these steps.```pythonimport itertools# Define the set S_4S_4 = {1, 2, 3, 4}# Function to calculate the capacity (product) of a setdef capacity(subset): if len(subset) == 0: return 0 prod = 1 for num in subset: prod *= num return prod# Generate all subsets of S_4all_subsets = list(itertools.chain.from_iterable(itertools.combinations(S_4, r) for r in range(len(S_4)+1)))# Calculate the sum of capacities of even subsetseven_capacity_sum = 0for subset in all_subsets: cap = capacity(subset) if cap % 2 == 0: even_capacity_sum += capprint(even_capacity_sum)``````output112```The sum of the capacities of all even subsets of ( S_4 = {1, 2, 3, 4} ) is (boxed{112}).The sum of the capacities of all even subsets of ( S_4 = {1, 2, 3, 4} ) is (boxed{112}).