answer:To determine how many kilometers Martin traveled during his eight-hour business trip, we can split the trip into two halves: the first half and the second half. Each half of the trip took four hours because:[ text{Total trip time} = 8 text{ hours} ][ text{Trip time for each half} = frac{8 text{ hours}}{2} = 4 text{ hours} ]Martin traveled for 4 hours at a speed of 70 kilometers per hour during the first half, and for the remaining 4 hours at a speed of 85 kilometers per hour during the second half.We can use the formula for distance:[ text{Distance} = text{Speed} times text{Time} ]Let's calculate the distances for each half of the trip and then sum them up to get the total distance.```python# Given valuestime_each_half = 4 # hours# Speeds in km/hspeed_first_half = 70speed_second_half = 85# Distance traveled during the first halfdistance_first_half = speed_first_half * time_each_half# Distance traveled during the second halfdistance_second_half = speed_second_half * time_each_half# Total distance traveledtotal_distance = distance_first_half + distance_second_halfprint(total_distance)``````output620```Martin traveled a total of (boxed{620}) kilometers during his eight-hour business trip.

answer:To solve the given problem, we need to determine the coefficients (a_0, a_1, a_2,) and (a_3) such that the equation (x^3 = a_0 + a_1(x-2) + a_2(x-2)^2 + a_3(x-2)^3) holds for any real number (x). We then need to find the value of (a_0 + a_1 + a_2 + a_3). Steps to solve the problem:1. Understand the Equation: The given equation (x^3 = a_0 + a_1(x-2) + a_2(x-2)^2 + a_3(x-2)^3) needs to hold for any real number (x). This means that the polynomial on the right side must be identical to (x^3) when expanded.2. Expand the Right-Hand Side: Expand the polynomial (a_0 + a_1(x-2) + a_2(x-2)^2 + a_3(x-2)^3) and collect like terms, equating the coefficients of corresponding powers of (x) from both sides of the equation.3. Determine Coefficients: Solve the system of equations derived from equating the coefficients to find (a_0, a_1, a_2, a_3).4. Compute (a_0 + a_1 + a_2 + a_3): Add these coefficients to find the final result.Let's implement this step-by-step process using Python and Sympy.```pythonimport sympy as sp# Define the variable xx = sp.symbols('x')# Define the polynomial on the right sidea0, a1, a2, a3 = sp.symbols('a0 a1 a2 a3')rhs = a0 + a1*(x-2) + a2*(x-2)2 + a3*(x-2)3# Expand the right-hand siderhs_expanded = sp.expand(rhs)# Define the left-hand side polynomiallhs = x3# Collect coefficients from both sideslhs_coeffs = sp.Poly(lhs, x).all_coeffs()rhs_coeffs = sp.Poly(rhs_expanded, x).all_coeffs()# Set up the system of equationsequations = [sp.Eq(lhs_coeffs[i], rhs_coeffs[i]) for i in range(4)]# Solve for the coefficients a0, a1, a2, a3solution = sp.solve(equations, (a0, a1, a2, a3))# Calculate a0 + a1 + a2 + a3result = solution[a0] + solution[a1] + solution[a2] + solution[a3]# Print the final resultprint(result)``````output27```The value of (a_0 + a_1 + a_2 + a_3) given the equation (x^3 = a_0 + a_1(x-2) + a_2(x-2)^2 + a_3(x-2)^3) for any real number (x) is (boxed{27}).

answer:To evaluate (log_8 2), we need to find the exponent (x) such that (8^x = 2).We know that (8) can be written as (2^3), so we can rewrite the equation as:[(2^3)^x = 2]This simplifies to:[2^{3x} = 2^1]For the bases (2) to be equal, their exponents must be equal:[3x = 1]Solving for (x):[x = frac{1}{3}]Thus, (log_8 2 = frac{1}{3}).Let's confirm this result using Python's logarithm functions. We'll use the change of base formula to verify:[log_b a = frac{log_c a}{log_c b}]In this case:[log_8 2 = frac{log_{10} 2}{log_{10} 8}]We'll compute this using Python.```pythonimport math# Calculate log_82 using change of base formulalog_8_2 = math.log(2, 10) / math.log(8, 10)print(log_8_2)``````output0.3333333333333333```The calculated value of (log_8 2) using Python's logarithm functions is approximately (0.3333), which confirms that (log_8 2 = frac{1}{3}).Therefore, the final answer is:[boxed{frac{1}{3}}]

answer:The relationship stated in the problem is that the force needed to loosen a bolt (F) varies inversely with the length of the handle (L). This can be expressed as:[ F = frac{k}{L} ]where (k) is a constant. Given that a 9-inch handle requires 375 pounds of force, we can use this information to find (k):[ 375 = frac{k}{9} ]From this, we can solve for (k).Once we have (k), we can use it to find the force required for a handle of length 15 inches:[ F_{15} = frac{k}{15} ]Let's solve this using Python code.```python# Given valuesforce_9_inches = 375length_9_inches = 9# Calculate the constant kk = force_9_inches * length_9_inches# Length of the new wrench handlelength_15_inches = 15# Calculate the new force requiredforce_15_inches = k / length_15_inchesprint(force_15_inches)``````output225.0```Using the calculations, we find that a wrench with a handle length of 15 inches will require ( boxed{225} ) pounds of force to loosen the same bolt.

answer:To find the limit of the function:[lim_{x rightarrow 0} frac{2x}{operatorname{tg}left(2pileft(x + frac{1}{2}right)right)}]we need to follow these steps:1. Set up the expression: Define the function inside the limit in terms of (x).2. Simplify the expression: Use trigonometric identities and simplifications to make the limit calculation easier.3. Compute the limit: Use SymPy's limit functionality to find the limit as (x) approaches 0.Let's implement this using Python and SymPy.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the function inside the limitfunction = (2 * x) / sp.tan(2 * sp.pi * (x + 1/2))# Compute the limit as x approaches 0limit_result = sp.limit(function, x, 0)print(limit_result)``````output1/pi```The limit of the function [lim_{x rightarrow 0} frac{2x}{operatorname{tg}left(2pileft(x + frac{1}{2}right)right)}]is (boxed{frac{1}{pi}}).

answer:To solve the problem, we need to find the value of (a_4 + a_5 + a_6) given that the first term (a_1 = 3) and the sum of the first three terms is 21. Here's how we can break it down:1. Define the Arithmetic Sequence: The (n)-th term of an arithmetic sequence can be written as: [ a_n = a_1 + (n-1)d ] where (a_1) is the first term and (d) is the common difference.2. Given Values and Equation: - The first term (a_1 = 3). - The sum of the first three terms is 21: [ a_1 + a_2 + a_3 = 21 ]3. Substitute the Given Values: Using the formula for the (n)-th term: [ a_2 = a_1 + d a_3 = a_1 + 2d ] Substituting these into the sum equation: [ a_1 + (a_1 + d) + (a_1 + 2d) = 21 ] Simplifying, we get: [ 3a_1 + 3d = 21 ] Since (a_1 = 3): [ 3 times 3 + 3d = 21 ] Solving for (d), we get: [ 9 + 3d = 21 3d = 12 d = 4 ]4. Find (a_4, a_5, a_6): Using the formula (a_n = a_1 + (n-1)d): [ a_4 = a_1 + 3d = 3 + 3 times 4 = 3 + 12 = 15 a_5 = a_1 + 4d = 3 + 4 times 4 = 3 + 16 = 19 a_6 = a_1 + 5d = 3 + 5 times 4 = 3 + 20 = 23 ]5. Calculate (a_4 + a_5 + a_6): [ a_4 + a_5 + a_6 = 15 + 19 + 23 ]Let's implement this in Python using sympy to ensure the correctness:```pythonimport sympy as sp# Define the first term and the sum equationa1 = 3sum_of_first_three = 21# Define the common difference 'd'd = sp.symbols('d')# Sum of the first three termseq = a1 + (a1 + d) + (a1 + 2*d) - sum_of_first_three# Solve for 'd'd_value = sp.solve(eq, d)[0]# Define the terms a4, a5, a6a4 = a1 + 3*d_valuea5 = a1 + 4*d_valuea6 = a1 + 5*d_value# Calculate sum of a4, a5, a6sum_next_three_terms = a4 + a5 + a6# Print the final resultprint(sum_next_three_terms)``````output57```The value of (a_4 + a_5 + a_6) is (boxed{57}).The sum of (a_4 + a_5 + a_6) in the arithmetic sequence, given that (a_1 = 3) and the sum of the first three terms is 21, is (boxed{57}).