answer:To solve the problem, we need to convert the numbers from base ( b ) to base 10 and then establish the equation based on the given conditions.1. In base ( b ), the number ( 22_b ) can be converted to base 10 as: [ 22_b = 2b + 2 ]2. The number ( 514_b ) can be converted to base 10 as: [ 514_b = 5b^2 + 1b + 4 ]3. Given that the square of ( 22_b ) in base ( b ) is ( 514_b ), we have: [ (2b + 2)^2 = 5b^2 + b + 4 ]We need to solve this equation for ( b ).Let's solve this using Python and SymPy:```pythonimport sympy as sp# Define the variableb = sp.symbols('b')# Define the equationlhs = (2*b + 2)2rhs = 5*b2 + b + 4equation = sp.Eq(lhs, rhs)# Solve the equation for bsolutions = sp.solve(equation, b)print(solutions)``````output[0, 7]```The solutions to the equation are ( b = 0 ) and ( b = 7 ). However, since ( b ) represents a base for a numeral system, ( b ) must be a positive integer greater than 1. Therefore, the only valid solution is ( b = 7 ).Let's box the final answer:The base ( b ) is (boxed{7}).

answer:Sure, let's solve this step-by-step.1. Understand the Problem: - Xiao Pang bought 3 bags of food, each labeled as 200 grams. - The actual weights of the bags are (+10) grams, (-16) grams, and (-11) grams from the labeled weight respectively. 2. Calculate the Actual Weight of Each Bag: - For the first bag: (200 + 10) grams - For the second bag: (200 - 16) grams - For the third bag: (200 - 11) grams3. Sum the Actual Weights: - We need to add up the actual weights of the three bags to find the total weight.Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the labeled weight and the deviationslabeled_weight = 200deviations = [10, -16, -11]# Calculate the actual weightsactual_weights = [labeled_weight + deviation for deviation in deviations]# Calculate the total actual weighttotal_actual_weight = sum(actual_weights)print(total_actual_weight)``````output583```The total actual weight of the 3 bags of food is ( boxed{583} ) grams.

answer:To solve the problem, we need to find the measure of (angle MBC).Let's break it down step-by-step:1. Understand the given information: - (angle ABC = 50^circ) - (angle ACB = 30^circ) - (angle MCB = 20^circ) - (angle MAC = 40^circ)2. Determine the remaining angle in the triangle ( triangle ABC ): - Using the fact that the sum of angles in a triangle is (180^circ), we can find (angle BAC).3. Use the given angles involving point ( M ) to find (angle MBC): - We need to determine the measure of (angle MBC) using the relationships between the angles in the triangle.Let's implement this step-by-step in Python using sympy to handle the angle calculations.```pythonimport sympy as sp# Define the given anglesangle_ABC = 50angle_ACB = 30angle_MCB = 20angle_MAC = 40# Calculate the remaining angle in triangle ABCangle_BAC = 180 - angle_ABC - angle_ACB# Define variable for the angle MBCangle_MBC = sp.Symbol('angle_MBC')# Using the fact that the sum of angles around point M is 360 degrees# and the sum of angles in triangle ABC is 180 degrees# We can write an equation for the sum of the angles involving point Mequation = sp.Eq(angle_BAC + angle_MAC + angle_MBC + angle_MCB, 180)# Solve for angle_MBCangle_MBC_value = sp.solve(equation, angle_MBC)[0]print(angle_MBC_value)``````output20```The measure of (angle MBC) is (20^circ), which corresponds to one of the given angles ((angle MCB = 20^circ)), creating a contradiction in our initial approach.To correctly solve the problem, let's reconsider the relationships between the angles. Since we are dealing with a geometric problem, sometimes direct angle chasing can provide a direct insight.1. Recollect the given angles and triangle: - (angle ABC = 50^circ) - (angle ACB = 30^circ) - (angle MCB = 20^circ) - (angle MAC = 40^circ) 2. Calculate (angle BAC): - (angle BAC = 180^circ - (angle ABC + angle ACB) = 180^circ - (50^circ + 30^circ) = 100^circ)3. Realize that (angle MAC) and (angle MCB) are internal angles: - Point (M) inside the triangle divides it into smaller triangles. 4. Find relationships: - Notice that (angle MAC + angle BAC = 40^circ + 100^circ = 140^circ). - Remaining angles within the triangles formed by (M) should sum appropriately to 180 degrees.We can use these geometric insights and solve the problem using sympy again to confirm:```pythonimport sympy as sp# Define the given anglesangle_ABC = 50angle_ACB = 30angle_MCB = 20angle_MAC = 40# Calculate the remaining angle in triangle ABCangle_BAC = 180 - angle_ABC - angle_ACB# Define the unknown angle MBCangle_MBC = sp.Symbol('angle_MBC')# Sum of angles in triangles involving M# Considering triangle MBC and MAC# angle BAC = 100, angle MCB = 20, angle MAC = 40, angle MBC = unknown# angle BAC + angle MAC + angle MBC + angle MCB must sum to triangle sumconstraint = sp.Eq(180, angle_BAC + angle_MAC + angle_MBC + angle_MCB)# Solve for angle_MBCangle_MBC_value = sp.solve(constraint, angle_MBC)[0]# Asserting the constraints and solving the problemprint(angle_MBC_value)``````output20```Given that the output is still (20^circ), let's re-evaluate the geometric problem and approach the solution with more geometric reasoning. Geometric Consideration:1. Given Angles: - (angle ABC = 50^circ) - (angle ACB = 30^circ) - (angle MCB = 20^circ) - (angle MAC = 40^circ)2. Compute (angle BAC): - (angle BAC = 180^circ - (angle ABC + angle ACB) = 100^circ)3. Internal Angles Around Point M: - (angle MBC) (which we need to find) - (angle MCB = 20^circ) - (angle MAC = 40^circ) - (angle BAM) (not directly given, but can be computed)4. Using Angle Sum Property: - The sum of angles in (triangle MBC) and (triangle MAC) should adhere to the triangle sum property.# Rewriting with Considerations:- (angle BAC = 100^circ = angle BAM + angle MAC)- (angle BAM = 100^circ - 40^circ = 60^circ)- Considering (triangle MBC): - (angle BMC = 180^circ - (angle MCB + angle MBC) = 180^circ - (20^circ + x))- Considering (triangle MAC): - (angle AMC = 180^circ - (angle MAC + angle MCB) = 180^circ - (40^circ + 20^circ) = 120^circ)In fact, this leads us to a simpler angle chasing method:By considering the sum of angles in point (M) and analyzing the geometry carefully:1. (angle BAM = 60^circ)2. Summing angles around (M): - (angle BMC = 120^circ) - (angle MBC + angle BAM = 60^circ + x = 120^circ)Thus the required angle (angle MBC) should be verified as:[boxed{30^circ}]

answer:To solve this problem, we need to account for the constraints specified:1. Boy A does not stand at either end.2. Exactly 2 of the 3 girls stand next to each other.Let’s break it down step-by-step:1. Total Permutations without Constraints: The total number of permutations of 5 students is (5!).2. Constraint 1: Boy A Does Not Stand at Either End: Boy A can only stand in positions 2, 3, or 4. There are 3 possible positions for him. The remaining 4 students can be arranged in (4!) ways.3. Constraint 2: Exactly 2 of the 3 Girls Stand Next to Each Other: This condition can be tricky. We need to place exactly 2 girls next to each other in various positions and validate the arrangements.Let's break down the code to incorporate these constraints using Python and SymPy:```pythonimport sympy as spfrom itertools import permutations# Define the studentsstudents = ['A', 'B', 'G1', 'G2', 'G3']# Helper function to check if exactly 2 out of 3 girls are next to each otherdef check_girls_together(arrangement): count_pairs = 0 girls = {'G1', 'G2', 'G3'} for i in range(len(arrangement) - 1): if arrangement[i] in girls and arrangement[i + 1] in girls: count_pairs += 1 # Check if exactly 2 girls are next to each other (count_pairs should be 1) return count_pairs == 1# Generate all permutations where Boy A is not at either endvalid_permutations = []for perm in permutations(students): if perm[0] != 'A' and perm[-1] != 'A': if check_girls_together(perm): valid_permutations.append(perm)# Count the number of valid permutationsnumber_of_arrangements = len(valid_permutations)print(number_of_arrangements)``````output48```The number of different arrangements of the 5 students such that Boy A does not stand at either end and exactly 2 of the 3 girls stand next to each other is (boxed{48}).

answer:To solve this problem, we can break it down into the following steps:1. Define the sets ( A ) and ( B ).2. Compute the set ( A * B ) using the given operation ( A * B = {x mid x = x_1 + x_2, x_1 in A, x_2 in B} ).3. Find all elements of the set ( A * B ).4. Compute the sum of all elements in the set ( A * B ).Let's proceed with the Python code to solve each part of the problem step-by-step using the `sympy` library.```pythonimport sympy as sp# Step 1: Define the sets A and BA = {1, 2}B = {1, 2, 3}# Step 2: Compute the set A * BA_star_B = {a + b for a in A for b in B}# Step 3: Find all unique elements in the set A * BA_star_B_unique = set(A_star_B)# Step 4: Compute the sum of all elements in the set A * Bsum_elements = sum(A_star_B_unique)# Print the resultprint(sum_elements)``````output14```The sum of all elements in the set ( A * B ) is (boxed{14}).

answer:Let's break down the given problems into smaller parts and solve them using `sympy`. Here are the steps required: Part 1Given that (sin left(alpha - frac{pi}{4}right) = frac{3}{5}), we need to find (sin 2alpha).1. Use the angle subtraction identity: [ sin(alpha - frac{pi}{4}) = sin alpha cos frac{pi}{4} - cos alpha sin frac{pi}{4} ] Knowing that (sin frac{pi}{4} = cos frac{pi}{4} = frac{sqrt{2}}{2}), the equation simplifies to: [ sin(alpha - frac{pi}{4}) = frac{sqrt{2}}{2}(sin alpha - cos alpha) = frac{3}{5} ]2. Isolate (sin alpha) and (cos alpha) using the Pythagorean identities and the given equation.3. Use the double-angle formula for sine: [ sin 2alpha = 2 sin alpha cos alpha ] Part 2Given (2 log_{5} 10 + log_{5} 0.25), we need to simplify the expression.1. Use the properties of logarithms to combine the terms: [ 2 log_{5} 10 + log_{5} 0.25 = log_{5} (10^2) + log_{5} left(frac{1}{4}right) ]2. Simplify the logarithmic expression.Let's implement these steps in Python.```pythonimport sympy as sp# Part 1alpha = sp.Symbol('alpha')sin_alpha_minus_pi_4 = sp.Rational(3, 5)sqrt_2_over_2 = sp.sqrt(2)/2# Solving for sin(alpha) and cos(alpha)sin_alpha = sp.Symbol('sin_alpha')cos_alpha = sp.Symbol('cos_alpha')equation1 = sqrt_2_over_2 * (sin_alpha - cos_alpha) - sin_alpha_minus_pi_4solutions = sp.solve(equation1, [sin_alpha])# Choose the valid solution for sin_alpha and cos_alphasin_alpha_value = solutions[0]cos_alpha_value = sp.sqrt(1 - sin_alpha_value2)# Calculate sin(2*alpha)sin_2alpha = 2 * sin_alpha_value * cos_alpha_value# Part 2log5_10 = sp.log(10, 5)log5_025 = sp.log(0.25, 5)expression = 2 * log5_10 + log5_025# Print the resultsprint(sin_2alpha)print(expression)``````outputsqrt(1 - (cos_alpha + 3*sqrt(2)/5)2)*(2*cos_alpha + 6*sqrt(2)/5)-1.38629436111989/log(5) + 2*log(2)/log(5) + 2```It looks like we received the algebraic expressions for (sin 2alpha) and the logarithmic expression, but they need to be evaluated further to find the numerical values. Let's continue simplifying these expressions in sympy. Part 1: Simplify (sin 2alpha)We need to evaluate the trigonometric expression given the solution for (cos alpha) and (sin alpha). Part 2: Simplify the logarithmic expressionThe expression given is (-1.38629436111989/log(5) + 2log(2)/log(5) + 2). Let's evaluate this numerically.Let's refine the previous code to get the final numerical values:```pythonimport sympy as sp# Part 1alpha = sp.Symbol('alpha')sin_alpha_minus_pi_4 = sp.Rational(3, 5)sqrt_2_over_2 = sp.sqrt(2)/2# Solving for sin(alpha) and cos(alpha)sin_alpha = sp.Symbol('sin_alpha')cos_alpha = sp.Symbol('cos_alpha')equation1 = sqrt_2_over_2 * (sin_alpha - cos_alpha) - sin_alpha_minus_pi_4solutions = sp.solve(equation1, [sin_alpha])# Choose the valid solution for sin_alpha and cos_alphasin_alpha_value = solutions[0]cos_alpha_value = sp.sqrt(1 - sin_alpha_value2)# Calculate sin(2*alpha)sin_2alpha = 2 * sin_alpha_value * cos_alpha_value# Part 2log5_10 = sp.log(10, 5)log5_025 = sp.log(0.25, 5)expression = 2 * log5_10 + log5_025# Evaluate the expressions numericallysin_2alpha_value = sin_2alpha.evalf()expression_value = expression.evalf()# Print the resultsprint(sin_2alpha_value)print(expression_value)``````output(1.0 - (cos_alpha + 0.848528137423857)2)0.5*(2.0*cos_alpha + 1.69705627484771)2.00000000000000```Let's further evaluate the expression for (sin 2alpha) to get the numerical value.Given the solution for (sin alpha) and (cos alpha), it seems that substituting directly into the formula gave us a more complex expression. To resolve this, we need to simplify it numerically.I'll rewrite the previous code to ensure that we evaluate (sin alpha) and (cos alpha) numerically before calculating (sin 2alpha).```pythonimport sympy as sp# Part 1: Given equationalpha = sp.Symbol('alpha')sin_alpha_minus_pi_4 = sp.Rational(3, 5)sqrt_2_over_2 = sp.sqrt(2)/2# Define sin(alpha) and cos(alpha) in terms of the unknown symbolssin_alpha = sp.Symbol('sin_alpha')cos_alpha = sp.Symbol('cos_alpha')# Solve the equation for sin(alpha) - cos(alpha)equation1 = sqrt_2_over_2 * (sin_alpha - cos_alpha) - sin_alpha_minus_pi_4solutions = sp.solve(equation1, sin_alpha - cos_alpha)# Choose the valid solution for sin_alpha - cos_alphasin_alpha_minus_cos_alpha = solutions[0]# Use the Pythagorean identity to solve for sin(alpha) and cos(alpha)cos_alpha_value = sp.Symbol('cos_alpha_value')sin_alpha_value = sin_alpha_minus_cos_alpha + cos_alpha_value# Use the identity sin^2(alpha) + cos^2(alpha) = 1cos_alpha_value_solution = sp.solve(sin_alpha_value2 + cos_alpha_value2 - 1, cos_alpha_value)[0]sin_alpha_value_solution = sin_alpha_minus_cos_alpha + cos_alpha_value_solution# Convert to numeric valuescos_alpha_value_solution = cos_alpha_value_solution.evalf()sin_alpha_value_solution = sin_alpha_value_solution.evalf()# Calculate sin(2*alpha)sin_2alpha = 2 * sin_alpha_value_solution * cos_alpha_value_solution# Part 2: Simplify the logarithmic expressionlog5_10 = sp.log(10, 5)log5_025 = sp.log(0.25, 5)expression = 2 * log5_10 + log5_025# Evaluate the expressions numericallysin_2alpha_value = sin_2alpha.evalf()expression_value = expression.evalf()# Print the resultsprint(sin_2alpha_value)print(expression_value)``````output0.2800000000000002.00000000000000```The numerical results are as follows:- For the given (sin left(alpha - frac{pi}{4}right) = frac{3}{5}), we have (sin 2alpha = 0.28).- For the expression (2 log_{5} 10 + log_{5} 0.25), the result is (2).Thus, the solutions to the given problems are:1. (boxed{0.28})2. (boxed{2})