answer:To determine the number of girls in the class, we'll use the given ratio and the total number of students. Here's the plan:1. Let ( B ) be the number of boys and ( G ) be the number of girls.2. According to the ratio 5:8, we have: [ frac{B}{G} = frac{5}{8} ]3. We are also given that the total number of students in the class is 260: [ B + G = 260 ]4. Use these equations to solve for ( G ).To solve this step-by-step:1. Express ( B ) in terms of ( G ) using the ratio: [ B = frac{5}{8}G ]2. Substitute ( B ) into the total student equation: [ frac{5}{8}G + G = 260 ]3. Then solve for ( G ).Let's perform the calculations using Python.```pythonfrom sympy import symbols, Eq, solve# Define the symbolsG = symbols('G')# Given ratio and total students in the classratio_b_to_g = 5 / 8total_students = 260# Express B in terms of G using the ratio B/G = 5/8B = ratio_b_to_g * G# Equation representing the total number of studentsequation = Eq(B + G, total_students)# Solve for G (number of girls)number_of_girls = solve(equation, G)[0]print(number_of_girls.evalf())``````output160.000000000000```The number of girls in the class is ( boxed{160} ).

answer:To determine the number of zeros of the quadratic function (y = x^2 + x - 1), we need to find the roots of the equation (x^2 + x - 1 = 0). The roots of a quadratic equation (ax^2 + bx + c = 0) can be found using the quadratic formula:[x = frac{-b pm sqrt{b^2 - 4ac}}{2a}]In this case, (a = 1), (b = 1), and (c = -1).Let's use Python and SymPy to find the roots of the given quadratic equation and determine the number of zeros.```pythonimport sympy as sp# Define the variable and the quadratic equationx = sp.symbols('x')quadratic_eq = x2 + x - 1# Find the roots of the quadratic equationroots = sp.solve(quadratic_eq, x)# The number of roots (zeros) of the quadratic functionnum_zeros = len(roots)print(num_zeros)``````output2```The number of zeros of the quadratic function (y = x^2 + x - 1) is (boxed{2}).

answer:To solve this problem, we need to calculate the different total prices that can be obtained by pairing each gift with each packaging box. The prices of gifts are: [2, 5, 8, 11, 14] yuan.The prices of packaging boxes are: [3, 5, 7, 9, 11] yuan.First, we need to compute the sum for every possible pair of gift and packaging box. Then, we will count the number of unique total prices.Let's solve this using Python code.Here's the plan:1. Generate all combinations of pairs (gift price, packaging box price).2. Calculate the total price for each pair.3. Store the total prices in a set to ensure uniqueness.4. Count the number of unique total prices.```python# Define the prices of gifts and packaging boxesgift_prices = [2, 5, 8, 11, 14]box_prices = [3, 5, 7, 9, 11]# Calculate all possible total prices by pairing each gift with each packaging boxtotal_prices = set()for gift in gift_prices: for box in box_prices: total_price = gift + box total_prices.add(total_price)# Calculate the number of unique total pricesnum_unique_total_prices = len(total_prices)print(num_unique_total_prices)``````output19```The number of different total prices that can be obtained by pairing each gift with each packaging box is (boxed{19}).

answer:To find the height of the pyramid (P-ABCD), we need to determine the perpendicular distance from point (P) to the plane defined by points (A), (B), and (D).Here are the steps to solve this problem:1. Compute the normal vector to the plane (ABD): The normal vector (mathbf{N}) to the plane can be found using the cross product of vectors (overrightarrow{AB}) and (overrightarrow{AD}).2. Find the equation of the plane: Using the normal vector (mathbf{N}) and point (A), we can find the equation of the plane.3. Calculate the perpendicular distance from point (P) to the plane: Use the formula for the distance from a point to a plane to find the height of the pyramid.Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the vectorsAB = sp.Matrix([4, -2, 3])AD = sp.Matrix([-4, 1, 0])AP = sp.Matrix([-6, 2, -8])# Step 1: Compute the normal vector to the plane ABDN = AB.cross(AD)# Step 2: Find the equation of the plane using point A and normal vector N# Plane equation: N1*x + N2*y + N3*z = DA = sp.Matrix([0, 0, 0]) # Point A is the originD = N.dot(A)# Step 3: Calculate the perpendicular distance from point P to the planeP = sp.Matrix([-6, 2, -8])distance = sp.Abs(N.dot(P) - D) / sp.sqrt(N.dot(N))# Output the height of the pyramidprint(distance)``````output2```The height of the pyramid (P-ABCD) is (boxed{2}).The height of the pyramid (P-ABCD) from point (P) to the plane (ABD) is (boxed{2}).

answer:To solve the problem, we need to calculate Esperanza's gross monthly salary based on her expenses and savings.Given:- Rent: 600- Food: (frac{3}{5}) of rent- Mortgage: 3 times the cost of food- Savings: 2000- Taxes: (frac{2}{5}) of savingsWe need to:1. Calculate the amount spent on food.2. Calculate the amount spent on mortgage.3. Calculate the taxes.4. Sum up all expenses and savings to get the gross monthly salary.Let's calculate each step one by one:1. Calculate the amount spent on food:[ text{Food} = frac{3}{5} times text{Rent} ]2. Calculate the amount spent on mortgage:[ text{Mortgage} = 3 times text{Food} ]3. Calculate the taxes:[ text{Taxes} = frac{2}{5} times text{Savings} ]4. Calculate the gross monthly salary:[ text{Gross Monthly Salary} = text{Rent} + text{Food} + text{Mortgage} + text{Savings} + text{Taxes} ]We will now implement this using Python to ensure the result is accurate.```python# Given valuesrent = 600savings = 2000# Calculate the amount spent on foodfood = (3/5) * rent# Calculate the amount spent on mortgagemortgage = 3 * food# Calculate the taxestaxes = (2/5) * savings# Calculate the gross monthly salarygross_monthly_salary = rent + food + mortgage + savings + taxesprint(gross_monthly_salary)``````output4840.0```Esperanza's gross monthly salary is (boxed{4840}).

answer:To solve this problem, we'll use properties of exponents and logarithms. Let's break down the solution step-by-step:1. Identify Known Exponents: We are given: [ a^m = 2, quad a^n = 4, quad a^k = 32 ]2. Express Exponents in Terms of Each Other: Convert the given exponential forms into base 2: [ a^m = 2^1, quad a^n = 2^2, quad a^k = 2^5 ] From this, we can set up the equations: [ m log_a a = log_a 2 implies m = log_a 2 ] [ n log_a a = log_a 4 implies n = log_a 4 = 2 log_a 2 ] [ k log_a a = log_a 32 implies k = log_a 32 = 5 log_a 2 ] Let ( log_a 2 = x ). Then: [ m = x, quad n = 2x, quad k = 5x ]3. Substitute and Simplify: We need to find ( a^{3m + 2n - k} ): [ 3m + 2n - k = 3x + 2(2x) - 5x = 3x + 4x - 5x = 2x ] Therefore: [ a^{3m + 2n - k} = a^{2x} ] Since ( a^x = 2 ), we have: [ a^{2x} = (a^x)^2 = 2^2 = 4 ]Let's write the Python code to verify these steps using sympy:```pythonimport sympy as sp# Define the variablesa, m, n, k = sp.symbols('a m n k')# Given expressionsexpr1 = am - 2expr2 = an - 4expr3 = ak - 32# Solve for m, n, k in terms of log_a(2)log_a_2 = sp.log(2, a)m_val = log_a_2n_val = 2 * log_a_2k_val = 5 * log_a_2# Calculate 3m + 2n - kresult_exponent = 3 * m_val + 2 * n_val - k_val# Calculate a^(3m + 2n - k)final_result = aresult_exponentsimplified_result = sp.simplify(final_result)# Substitute a^x = 2simplified_result_substitute = simplified_result.subs(alog_a_2, 2)# Print the final resultprint(simplified_result_substitute)``````output4```The value of (a^{3m + 2n - k}) given (a^m = 2), (a^n = 4), and (a^k = 32) is (boxed{4}).