answer:To solve the problem, we need to calculate the following expression:[sqrt{27} times sqrt{frac{8}{3}} div sqrt{frac{1}{2}}]Let's break it down into manageable steps:1. Compute (sqrt{27}).2. Compute (sqrt{frac{8}{3}}).3. Compute (sqrt{frac{1}{2}}).4. Perform the multiplication and division operations as specified.Let's implement this in Python using the `sympy` library to ensure accurate symbolic computation.```pythonimport sympy as sp# Define the expression componentssqrt_27 = sp.sqrt(27)sqrt_8_3 = sp.sqrt(sp.Rational(8, 3))sqrt_1_2 = sp.sqrt(sp.Rational(1, 2))# Compute the expressionresult = sqrt_27 * sqrt_8_3 / sqrt_1_2# Simplify the resultsimplified_result = sp.simplify(result)print(simplified_result)``````output12```The result of the expression (sqrt{27} times sqrt{frac{8}{3}} div sqrt{frac{1}{2}}) is (boxed{12}).

answer:To solve the problem, we need to define the new operation (oplus) and use it within the function (f(x) = (1 oplus x)x - (2 oplus x)). We will then find the maximum value of (f(x)) over the interval (x in [-2, 2]).Here are the steps:1. Define the operation (oplus).2. Define the function (f(x)).3. Compute (f(x)) for different cases based on the definition of (oplus).4. Find the maximum value of (f(x)) in the interval ([-2, 2]).Let's implement this step-by-step using Python and sympy:```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the op operationdef op(a, b): if a >= b: return a else: return b2# Define the function f(x) = (1 ⊕ x)x - (2 ⊕ x)def f(x): return (op(1, x) * x) - op(2, x)# Calculate f(x)f_expr = f(x)# Find the derivative of f(x)f_prime = sp.diff(f_expr, x)# Solve for critical points in the interval [-2, 2]critical_points = sp.solve(f_prime, x)critical_points = [point for point in critical_points if point.is_real and -2 <= point <= 2]# Include the endpoints of the interval as potential maximum pointscritical_points.extend([-2, 2])# Evaluate f(x) at each critical pointmax_value = max(f(x).evalf(subs={x: point}) for point in critical_points)print(max_value)``````outputTraceback (most recent call last): File " f_expr = f(x) File " return (op(1, x) * x) - op(2, x) File " if a >= b:TypeError: cannot determine truth value of Relational```The error occurs because the `op` function cannot directly compare symbolic expressions with a relational operator like `>=`. We need to handle these cases by defining the function such that it can evaluate the conditions based on the value of (x).We can redefine the `op(a, b)` function to handle symbolic expressions using conditional expressions. This will allow us to properly evaluate (f(x)) and find its maximum value.Here's the modified code:```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the op operation using Piecewisedef op(a, b): return sp.Piecewise((a, a >= b), (b2, a < b))# Define the function f(x) = (1 ⊕ x)x - (2 ⊕ x)def f(x): return (op(1, x) * x) - op(2, x)# Calculate f(x)f_expr = f(x)# Find the derivative of f(x)f_prime = sp.diff(f_expr, x)# Solve for critical points in the interval [-2, 2]critical_points = sp.solveset(f_prime, x, domain=sp.Interval(-2, 2))# Evaluate f(x) at each critical point and at the endpointscandidates = [point for point in critical_points] + [-2, 2]# Calculate the function value at each candidate pointvalues = [f_expr.subs(x, point).evalf() for point in candidates]# Find the maximum valuemax_value = max(values)print(max_value)``````output6.00000000000000```The maximum value of the function (f(x) = (1 oplus x)x - (2 oplus x)) over the interval (x in [-2, 2]) is (boxed{6}).

answer:To solve this problem, we need to determine the number of male attendees. We are given:1. The total number of attendees is 120.2. There are 4 more male attendees than female attendees.Let's use the following steps to solve this:1. Let ( F ) be the number of female attendees.2. Then, the number of male attendees ( M ) can be expressed as ( M = F + 4 ).3. The total number of attendees is the sum of male and female attendees.So we get the equation:[ F + M = 120 ]By substituting ( M ) from the second step into the equation, we get:[ F + (F + 4) = 120 ]We can solve this equation to find the value of ( F ), and then use it to find ( M ).Let's do the calculations in Python.```pythonfrom sympy import symbols, Eq, solve # Define the variables F = symbols('F') # Define the equation F + (F + 4) = 120 equation = Eq(F + (F + 4), 120) # Solve the equation female_attendees = solve(equation, F)[0] # Calculate the number of male attendees male_attendees = female_attendees + 4 print((female_attendees, male_attendees))``````output(58, 62)```The calculation shows that there are 58 female attendees and 62 male attendees.So, the number of male attendees is (boxed{62}).

answer:We need to determine the sequence defined by the conditions:1. ( a_{1} = a_{2} = 1 ),2. ( a_{3} = 2 ),3. ( a_{n} cdot a_{n+1} cdot a_{n+2} neq 1 ),4. ( a_{n} cdot a_{n+1} cdot a_{n+2} cdot a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3} ).Given these constraints, we need to find the sum ( a_{1} + a_{2} + cdots + a_{100} ).To solve this, we first need to identify and generate the sequence ( a_n ) that adheres to the given conditions. Based on the conditions given:- The product of any three consecutive terms is not equal to 1.- The product of any four consecutive terms equals their sum.We'll start by calculating and iterating over the sequence up to ( a_{100} ).Let's write the necessary Python code to generate the sequence and calculate the sum of the first 100 terms.```python# Initializing the first three termsa = [1, 1, 2]# Define the relationship to calculate subsequent termsdef next_term(a): n = len(a) if n < 3: raise ValueError("List must have at least 3 elements to proceed.") # We solve for a_{n+3} using the formula a_n * a_{n+1} * a_{n+2} * a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} # Rearranging the formula: a_{n+3} = (a_n + a_{n+1} + a_{n+2}) / (a_n * a_{n+1} * a_{n+2} - 1) numerator = a[-3] + a[-2] + a[-1] denominator = a[-3] * a[-2] * a[-1] - 1 if denominator == 0: raise ValueError("Denominator became zero, invalid sequence.") return numerator / denominator# Generate the sequence up to a_{100}while len(a) < 100: a.append(next_term(a))# Calculate the sum of the first 100 termssum_first_100_terms = sum(a)print(sum_first_100_terms)``````output200.0```The sum of the first 100 terms of the sequence is (boxed{200}).

answer:To solve this problem, we need to use the properties of triangles and trigonometric identities. Here's a step-by-step approach:1. Identify known values and variables: - (angle A = 60^circ) - (AB = c = 2) - Area of (triangle ABC = frac{sqrt{3}}{2}) - Let (AC = b) - Let (BC = a)2. Use the area formula for a triangle: The area of a triangle can be given by: [ text{Area} = frac{1}{2} times AB times AC times sin(angle A) ] Substituting the known values: [ frac{sqrt{3}}{2} = frac{1}{2} times 2 times b times sin(60^circ) ]3. Simplify the equation: [ frac{sqrt{3}}{2} = frac{1}{2} times 2 times b times frac{sqrt{3}}{2} ] [ frac{sqrt{3}}{2} = b times frac{sqrt{3}}{2} ]4. Solve for (b): By dividing both sides by (frac{sqrt{3}}{2}), we get: [ b = 1 ]Let's verify this calculation with Python and sympy to ensure accuracy.```pythonimport sympy as sp# Given valuesangle_A = 60 # degreesAB = 2area = sp.Rational(1, 2) * sp.sqrt(3)# Define AC as a variableAC = sp.symbols('AC')# Sin of 60 degreessin_60 = sp.sin(sp.rad(60))# Area formula: 1/2 * AB * AC * sin(angle_A)area_formula = (1 / 2) * AB * AC * sin_60# Equation: area_formula = given areaequation = sp.Eq(area_formula, area)# Solve for ACsolution = sp.solve(equation, AC)[0]print(solution)``````output1.00000000000000```The value of ( |AC| ) has been confirmed to be 1. Therefore, the length of ( |AC| ) is (boxed{1}).

answer:To solve this problem, we need to consider the total number of permutations (arrangements) of the students and then the number of permutations where students A and B are next to each other. Steps:1. Total Permutations Without Any Restriction: Calculate the total number of permutations of 6 students. 2. Permutations With A and B Together: Treat students A and B as a single entity or "block." This reduces the problem to arranging 5 entities (the A-B block and the other 4 students). 3. Internal Permutations of the A-B Block: Since A and B can switch places within their block, multiply the number of permutations by 2 to account for this. Calculation:1. Total permutations of 6 students is (6!).2. When A and B are treated as a single block, the number of permutations is (5!).3. Multiply the result by 2 to account for the internal arrangement of A and B within their block.Let's implement this in Python using the sympy library to ensure the accuracy of the calculations.```pythonimport sympy as sp# Total number of studentsn = 6# Total permutations of 6 studentstotal_permutations = sp.factorial(n)# Treat A and B as a single entity (block), so we have 5 entities to arrangeab_block_permutations = sp.factorial(n - 1) # This is 5!# Since A and B can switch places within their block, multiply by 2ab_permutations = ab_block_permutations * 2# Print the resultprint(ab_permutations)``````output240```The number of arrangements where students A and B are next to each other among the 6 students is (boxed{240}).To break it down:- Total permutations of 6 students: (6! = 720)- Treat A and B as a single entity, reducing the problem to 5 entities: (5! = 120)- Since A and B can switch places within their block: (120 times 2 = 240)Thus, there are (boxed{240}) valid arrangements where A and B are next to each other.