answer:To determine how many gallons of milk Aunt May has left, we'll start by calculating the total amount of milk she collected today, add any milk left over from the previous day, and then subtract the amount she sold.Let's break down the steps:1. Calculate the total milk collected today: - Morning: 365 gallons - Evening: 380 gallons2. Add the milk left over from yesterday.3. Subtract the amount of milk sold to the local ice cream factory.Finally, we'll use Python to iterate through these calculations to ensure accuracy.Here's the Python code that implements these steps:```python# Given datamorning_milk = 365evening_milk = 380sold_milk = 612leftover_milk_from_yesterday = 15# Step 1: Calculate the total milk collected todaytotal_milk_collected_today = morning_milk + evening_milk# Step 2: Add the milk left over from yesterdaytotal_milk_available = total_milk_collected_today + leftover_milk_from_yesterday# Step 3: Subtract the amount of milk soldmilk_left = total_milk_available - sold_milkprint(milk_left)``````output148```The final amount of milk Aunt May has left is:(boxed{148}) gallons.

answer:To solve the problem, we need to consider the following points:1. The number is even, so its units digit must be one of 0, 2, 4, 6, or 8.2. The sum of its digits is 89, which is an odd number.Given that the sum of the digits is odd, we can deduce that the units digit must be an odd number because the sum of an even number of even digits (which includes the units digit) will be even. So, the sum of the digits being odd means the units digit must be odd.The only odd digits that make the number even (because the units digit must be even) are 0, 2, 4, 6, and 8. The only even digits from these with the sum being odd conditions are 2, 6, and 8. So, let's analyze each of these possible units digits:- If the units digit is 2, then the sum of the other nine digits must be 87.- If the units digit is 6, then the sum of the other nine digits must be 83.- If the units digit is 8, then the sum of the other nine digits must be 81.We'll use Python to verify which of these conditions can be met to form a valid 10-digit number.```pythondef check_units_digit(): for units_digit in [2, 6, 8]: sum_of_other_digits = 89 - units_digit # Check if sum_of_other_digits can be split into 9 digits (0-9) if sum_of_other_digits <= 81: # max sum of 9 digits (each 9) = 81 return units_digit return Noneunits_digit = check_units_digit()print(units_digit)``````output8```Based on the output, the units digit of the 10-digit even number whose digits sum to 89 is (boxed{8}).Thus, the correct answer is:E) 8

answer:To solve the integral (int_0^{pi/2} frac{1}{1 + tan(sqrt{2} x)} , dx), we will use the symbolic computation capabilities of SymPy to find the exact value.Let's break it down into steps:1. Define the function ( f(x) = frac{1}{1 + tan(sqrt{2} x)} ).2. Set up the definite integral from 0 to (pi/2).3. Use SymPy to compute the integral.Here is the Python code to perform these steps:```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the function f(x)f_x = 1 / (1 + sp.tan(sp.sqrt(2) * x))# Define the integralintegral = sp.integrate(f_x, (x, 0, sp.pi/2))# Display the resultprint(integral)``````output-sqrt(2)*log(1 + tan(sqrt(2)*pi/2)2)/8 + pi/4 + sqrt(2)*(log(-1 - tan(sqrt(2)*pi/2)) + I*pi)/4```The output provided by SymPy indicates a complex expression, which involves logarithms and complex numbers. Given the trigonometric function inside the integral, it is possible that the symbolic computation yields a form that can be simplified further or has additional properties.To make sense of the output, let's attempt to simplify the expression manually. For definite integrals over standard intervals involving trigonometric functions, sometimes symmetries or special properties simplify the result.We will try another approach:- Use the symmetry properties or substitution that might simplify the integral.- Simplify the result to get the final answer.Let’s reconsider the integral and use a trigonometric identity or substitution that might simplify this integral. Notice that:- (tanleft(frac{pi}{2} - thetaright) = cot(theta))- This symmetry might help us simplify the integral.Consider the substitution (u = frac{pi}{2} - x), then (du = -dx).Let's rewrite the integral with this substitution to see if it simplifies:[int_0^{pi/2} frac{1}{1 + tan(sqrt{2} x)} , dx = int_{pi/2}^0 frac{1}{1 + tan(sqrt{2} (frac{pi}{2} - u))} , (-du) ]Using (tan(frac{pi}{2} - u) = cot(u)):[= int_0^{pi/2} frac{1}{1 + cot(sqrt{2} u)} , du]Since (cot(u) = frac{1}{tan(u)}):[= int_0^{pi/2} frac{1}{1 + frac{1}{tan(sqrt{2} u)}} , du= int_0^{pi/2} frac{tan(sqrt{2} u)}{tan(sqrt{2} u) + 1} , du]Summing the integrals from both expressions, we get:[int_0^{pi/2} frac{1}{1 + tan(sqrt{2} x)} , dx + int_0^{pi/2} frac{tan(sqrt{2} u)}{tan(sqrt{2} u) + 1} , du = int_0^{pi/2} 1 , dx]Since (int_0^{pi/2} 1 , dx = frac{pi}{2}), we get:[int_0^{pi/2} frac{1}{1 + tan(sqrt{2} x)} , dx + int_0^{pi/2} frac{tan(sqrt{2} x)}{1 + tan(sqrt{2} x)} , dx = frac{pi}{2}]These two integrals sum to (frac{pi}{2}), which gives us:[int_0^{pi/2} frac{1}{1 + tan(sqrt{2} x)} , dx = int_0^{pi/2} frac{tan(sqrt{2} x)}{1 + tan(sqrt{2} x)} , dx = frac{pi}{4}]Thus, the integral is:[boxed{frac{pi}{4}}]

answer:To solve this problem, let's break it down into steps:1. Understand the Problem: Each operation of the vacuum pump removes 50% of the air from the container. This means that after each operation, only 50% of the air remains. 2. Formulate the Problem: Let ( n ) be the number of operations. After ( n ) operations, the fraction of air remaining is ( (0.5)^n ). 3. Set Up the Inequality: We need to find the smallest integer ( n ) such that ( (0.5)^n < 0.001 ) (since 0.1% is equivalent to 0.001).4. Solve the Inequality: Logarithmic functions will help solve this inequality. Using the given logarithm values, we can solve for ( n ): [ log_{10}(0.5^n) < log_{10}(0.001) ] [ n log_{10}(0.5) < log_{10}(0.001) ] Given that ( log_{10}(2) = 0.3010 ), we know: [ log_{10}(0.5) = log_{10}(1/2) = -log_{10}(2) = -0.3010 ] So we have: [ n cdot (-0.3010) < -3 ] [ n > frac{3}{0.3010} ]5. Calculate the Value of ( n ):Let's implement these steps in Python to find the smallest integer ( n ).```pythonimport math# Given log valueslog_2 = 0.3010# Calculate the number of operationsn = math.ceil(3 / log_2)print(n)``````output10```The smallest integer ( n ) that satisfies the inequality ( (0.5)^n < 0.001 ) is (boxed{10}). This means the vacuum pump must operate 10 times to reduce the air inside the container to less than 0.1% of the original amount.

answer:Let's define the amounts given by the donors as follows:- Let ( x ) be the amount given by the first donor.- The second donor gave twice as much as the first, so the amount given by the second donor is ( 2x ).- The third donor gave three times as much as the second, so the amount given by the third donor is ( 3 times 2x = 6x ).- The fourth donor gave four times as much as the third, so the amount given by the fourth donor is ( 4 times 6x = 24x ).Given that the total donation is 132, we can write the equation as:[ x + 2x + 6x + 24x = 132 ]Let's solve this equation for ( x ).```pythonfrom sympy import symbols, Eq, solve# Define the variable and the equationx = symbols('x')equation = Eq(x + 2*x + 6*x + 24*x, 132)# Solve the equationsolution = solve(equation, x)print(solution)``````output[4]```The solution to the equation is ( x = 4 ).Thus, the first donor gave (boxed{4}) units.

answer:To solve this problem, we need to find ( a_{4} b_{4} ) given the system of equations and the known value ( a_{2} b_{3} = 7 ). Let's approach this step-by-step:1. Read and Understand the Equations: [ begin{cases} a_{1} b_{1} + a_{2} b_{3} = 1 a_{1} b_{2} + a_{2} b_{4} = 0 a_{3} b_{1} + a_{4} b_{3} = 0 a_{3} b_{2} + a_{4} b_{4} = 1 end{cases} ]2. Substitute the Known Value: Since ( a_{2} b_{3} = 7 ), we can substitute it into the first equation: [ a_{1} b_{1} + 7 = 1 implies a_{1} b_{1} = -6 ]3. Express Variables in Terms of Others: With the known equations, try finding expressions for ( a_1, a_2, a_3, a_4 ) and ( b_1, b_2, b_3, b_4 ).4. Solve for ( a_4 b_4 ): Use the equations to isolate and solve for ( a_{4} b_{4} ).Let's implement this step by step using Python with SymPy.```pythonimport sympy as sp# Define variablesa1, a2, a3, a4, b1, b2, b3, b4 = sp.symbols('a1 a2 a3 a4 b1 b2 b3 b4')# Define the equationseq1 = sp.Eq(a1 * b1 + a2 * b3, 1)eq2 = sp.Eq(a1 * b2 + a2 * b4, 0)eq3 = sp.Eq(a3 * b1 + a4 * b3, 0)eq4 = sp.Eq(a3 * b2 + a4 * b4, 1)# Known valueknown_eq = sp.Eq(a2 * b3, 7)# Substitute known value into the equationseq1 = eq1.subs(a2 * b3, 7)eq3 = eq3.subs(a2 * b3, 7)# Solve for unknownssolutions = sp.solve((eq1, eq2, eq3, eq4, known_eq), (a1, a2, a3, a4, b1, b2, b3, b4))# Get the value for a4 * b4a4 = solutions[a4]b4 = solutions[b4]a4_b4 = a4 * b4print(a4_b4)``````outputTraceback (most recent call last): File " a4 = solutions[a4]TypeError: list indices must be integers or slices, not Symbol```It seems that the output of the `sp.solve` is a list of dictionaries or tuples, and we should access the values accordingly. Let me correct the code to handle this properly.We will:1. Solve the equations.2. Extract the dictionary from the solution list.3. Compute the value of ( a_4 times b_4 ) using the solution dictionary.Let's update the code accordingly:```pythonimport sympy as sp# Define variablesa1, a2, a3, a4, b1, b2, b3, b4 = sp.symbols('a1 a2 a3 a4 b1 b2 b3 b4')# Define the equationseq1 = sp.Eq(a1 * b1 + a2 * b3, 1)eq2 = sp.Eq(a1 * b2 + a2 * b4, 0)eq3 = sp.Eq(a3 * b1 + a4 * b3, 0)eq4 = sp.Eq(a3 * b2 + a4 * b4, 1)# Known valueknown_eq = sp.Eq(a2 * b3, 7)# Substitute known value into the equationseq1 = eq1.subs(a2 * b3, 7)eq3 = eq3.subs(a2 * b3, 7)# Solve for unknownssolutions = sp.solve((eq1, eq2, eq3, eq4), (a1, a2, a3, a4, b1, b2, b3, b4))# Display solutionssolution_dict = solutions[0]# Extract the values for a4 and b4a4_value = solution_dict[a4]b4_value = solution_dict[b4]# Calculate a4 * b4a4_b4 = a4_value * b4_valueprint(a4_b4)``````outputTraceback (most recent call last): File " a4_value = solution_dict[a4]TypeError: tuple indices must be integers or slices, not Symbol```It appears that the solutions are being returned as a list of tuples instead of a list of dictionaries. Let's fix this by handling the output properly. We should extract the first tuple from the list and then access the elements of the tuple by position.Here is the corrected code:```pythonimport sympy as sp# Define variablesa1, a2, a3, a4, b1, b2, b3, b4 = sp.symbols('a1 a2 a3 a4 b1 b2 b3 b4')# Define the equationseq1 = sp.Eq(a1 * b1 + a2 * b3, 1)eq2 = sp.Eq(a1 * b2 + a2 * b4, 0)eq3 = sp.Eq(a3 * b1 + a4 * b3, 0)eq4 = sp.Eq(a3 * b2 + a4 * b4, 1)# Known valueknown_eq = sp.Eq(a2 * b3, 7)# Substitute known value into the equationseq1 = eq1.subs(a2 * b3, 7)eq3 = eq3.subs(a2 * b3, 7)# Solve for unknownssolutions = sp.solve((eq1, eq2, eq3, eq4, known_eq), (a1, a2, a3, a4, b1, b2, b3, b4))# Extract the first solution (if multiple solutions exist, they should all be equivalent)solution = solutions[0]# Assign the corresponding variables from the solution tuplea1_value, a2_value, a3_value, a4_value, b1_value, b2_value, b3_value, b4_value = solution# Calculate a4 * b4a4_b4 = a4_value * b4_valueprint(a4_b4)``````output-6```The value of (a_4 b_4) is (boxed{-6}).