answer:To solve the problem, we need to consider that the buses are operating on a circular route and traveling at the same speed. If there are two buses with a 21-minute interval, we need to determine the interval between buses when there are three buses.Here are the steps to solve the problem:1. Calculate the total time it takes for one bus to complete the circular route.2. Determine the interval when there are 3 buses.If the interval between two buses is 21 minutes with two buses, this means the total time for the route is 21 * 2 = 42 minutes. This is because having two buses with a 21-minute gap will cover the full circular route.For three buses on the same route:- The total route time is still 42 minutes.- The interval between each bus will be the total route time divided by the number of buses.Let's write the Python code to compute the new interval.```python# Total time for each bus to complete the circular routetotal_time_for_route = 21 * 2 # 42 minutes# Number of busesnumber_of_buses = 3# Interval between each bus when there are 3 busesnew_interval = total_time_for_route / number_of_busesprint(new_interval)``````output14.0```The interval between each bus when there are 3 buses operating on the same circular route, traveling at the same speed, will be (boxed{14}) minutes.

answer:The given equation is:[(x - 3)^4 + (x - 5)^4 = -8]Since ((x - 3)^4) and ((x - 5)^4) both represent fourth power terms, they are always non-negative for any real number (x). Therefore, the sum of two non-negative numbers can never equal a negative number like (-8). This implies that there are no real solutions to this equation.To make sure of our reasoning and to cover all possibilities including complex numbers, let's solve the equation in Python using numerical methods. We'll use the `sympy` library for symbolic computation to find all possible solutions.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the equationequation = (x - 3)4 + (x - 5)4 + 8# Solve the equationsolutions = sp.solve(equation, x)print(solutions)``````output[4 - I, 4 + I, 4 - sqrt(5)*I, 4 + sqrt(5)*I]```The equation ((x - 3)^4 + (x - 5)^4 = -8) has the following solutions in the complex plane:[4 - i, 4 + i, 4 - sqrt{5}i, 4 + sqrt{5}i]Hence, the solutions are:[boxed{4 - i, 4 + i, 4 - sqrt{5}i, 4 + sqrt{5}i}]

answer:Let's break down the problem and solve it step-by-step:1. Understanding the Problem: - We have a curve described by ( y = x^n ). - There is a tangent line to this curve at the point ((1, 0)). - This tangent line is parallel to the line ( 2x - y + 1 = 0 ).2. Determining the Slope of the Line: - The given line ( 2x - y + 1 = 0 ) can be rewritten in slope-intercept form as ( y = 2x + 1 ). - The slope of this line is ( 2 ).3. Finding the Slope of the Tangent Line: - The tangent line to the curve ( y = x^n ) at any point ( (x, y) ) has a slope given by the derivative ( frac{dy}{dx} ). - Evaluate the derivative at ( x = 1 ) to find the slope.4. Setting the Slope Condition: - The slope of the tangent line at ( x = 1 ) must equal the slope of the given line, which is ( 2 ).5. Solving for ( n ): - Derive the equation and solve for ( n ).Let's implement this in Python using SymPy.```pythonimport sympy as sp# Define the symbol n and the function y = x^nn = sp.symbols('n')x = sp.symbols('x')y = xn# Compute the derivative of y with respect to xdy_dx = sp.diff(y, x)# Evaluate the derivative at the point x = 1slope_at_point = dy_dx.subs(x, 1)# Set the slope equal to the given slope of 2 and solve for nslope_condition = sp.Eq(slope_at_point, 2)solution = sp.solve(slope_condition, n)print(solution)``````output[2]```The value of the real number ( n ) that satisfies the conditions given in the problem is (boxed{2}).The solution to the problem is that the value of the real number ( n ) is (boxed{2}).This confirms that the tangent line to the curve ( y = x^n ) at the point ((1, 0)) is indeed parallel to the line ( 2x - y + 1 = 0 ), when ( n = 2 ).

answer:The goal is to find the angle ( angle ABC ) where ( A = (-4,0,6) ), ( B = (-5,-1,2) ), and ( C = (-6,-1,3) ). To do this, we can use vector algebra. Here's a step-by-step plan:1. Compute the vectors ( overrightarrow{BA} ) and ( overrightarrow{BC} ): [ overrightarrow{BA} = A - B ] [ overrightarrow{BC} = C - B ]2. Compute the dot product of ( overrightarrow{BA} ) and ( overrightarrow{BC} ): [ overrightarrow{BA} cdot overrightarrow{BC} ]3. Compute the magnitudes of ( overrightarrow{BA} ) and ( overrightarrow{BC} ):4. Use the dot product and magnitudes to find the cosine of the angle between ( overrightarrow{BA} ) and ( overrightarrow{BC} ): [ cos theta = frac{overrightarrow{BA} cdot overrightarrow{BC}}{|overrightarrow{BA}| cdot |overrightarrow{BC}|} ]5. Find the angle ( theta ) in radians using ( arccos ) and then convert it to degrees.Let's implement this in Python to ensure accuracy.```pythonimport numpy as np# Define points A, B, and CA = np.array([-4, 0, 6])B = np.array([-5, -1, 2])C = np.array([-6, -1, 3])# Compute vectors BA and BCBA = A - BBC = C - B# Compute the dot product of BA and BCdot_product = np.dot(BA, BC)# Compute the magnitudes of BA and BCmagnitude_BA = np.linalg.norm(BA)magnitude_BC = np.linalg.norm(BC)# Compute the cosine of the anglecos_theta = dot_product / (magnitude_BA * magnitude_BC)# Compute the angle in radians, then convert to degreesangle_radians = np.arccos(cos_theta)angle_degrees = np.degrees(angle_radians)print(angle_degrees)``````output59.99999999999999```The angle ( angle ABC ) is approximately (boxed{60}) degrees.This value is computed using vector algebra and confirms that the result is accurate.

answer:To find the spherical distance between two points on the surface of the Earth given their latitude and longitude, we can use the Haversine formula. This formula calculates the great-circle distance between two points on a sphere given their longitudes and latitudes.Here's how to break down and solve the problem:1. Understand the Given Information: - Radius of the Earth ( R ) = 6371 km. - Coordinates of Shanghai: ((phi_1 = 31^{circ}, lambda_1 = 121^{circ})) - Coordinates of Taipei: ((phi_2 = 25^{circ}, lambda_2 = 121^{circ}))2. Convert Degrees to Radians: Latitude and longitude values are given in degrees, but trigonometric functions in Python use radians. Therefore, we'll convert these values to radians.3. Haversine Formula: The Haversine formula for the distance ( d ) between two points ((phi_1, lambda_1)) and ((phi_2, lambda_2)) is: [ a = sin^2left(frac{Delta phi}{2}right) + cos(phi_1) cos(phi_2) sin^2left(frac{Delta lambda}{2}right) ] [ c = 2 arctan2(sqrt{a}, sqrt{1-a}) ] [ d = R cdot c ] where: - (Delta phi = phi_2 - phi_1) - (Delta lambda = lambda_2 - lambda_1) - (R) is the radius of the Earth4. Calculate the Distance: Implement this formula in Python using the sympy and math libraries.Let's write the code to perform these calculations:```pythonimport sympy as spimport math# Define the radius of the EarthR = 6371 # in kilometers# Coordinates of Shanghai and Taipei in degreeslat_shanghai_deg = 31lon_shanghai_deg = 121lat_taipei_deg = 25lon_taipei_deg = 121# Convert degrees to radianslat_shanghai_rad = math.radians(lat_shanghai_deg)lon_shanghai_rad = math.radians(lon_shanghai_deg)lat_taipei_rad = math.radians(lat_taipei_deg)lon_taipei_rad = math.radians(lon_taipei_deg)# Calculate differences in coordinatesdelta_lat = lat_taipei_rad - lat_shanghai_raddelta_lon = lon_taipei_rad - lon_shanghai_rad# Haversine formulaa = (math.sin(delta_lat / 2) 2 + math.cos(lat_shanghai_rad) * math.cos(lat_taipei_rad) * math.sin(delta_lon / 2) 2)c = 2 * math.atan2(math.sqrt(a), math.sqrt(1 - a))distance = R * c# Round the result to the nearest kilometerdistance_rounded = round(distance)print(distance_rounded)``````output667```The spherical distance between Shanghai and Taipei, given their coordinates and the radius of the Earth, is approximately (boxed{667}) kilometers, rounded to the nearest kilometer.

answer:To solve this problem, we need to break it down into clear steps and use the information provided in the problem. Let's start by outlining the problem statement mathematically and then solve it using Python.Problem Breakdown:1. Initial Condition: - Two years ago, there were 20 trailer homes. - The average age of these homes was 18 years.2. New Homes Added: - ( x ) new trailer homes were added two years ago. - These new homes were brand new, meaning their age was 0 years at that time.3. Current Condition: - Today, the average age of all the trailer homes is 14 years.We need to determine the number of new trailer homes added (( x )).Mathematical Formulation:1. Total age of the original 20 trailer homes two years ago: [ text{Total age (two years ago)} = 20 times 18 = 360 text{ years} ]2. Age two years ago of the new ( x ) trailer homes is 0 (since they were brand new).3. Total number of trailer homes now: [ 20 + x ]4. Total age of all trailer homes now (taking into account that all ages have increased by two years): - Original 20 trailer homes: ( 360 + 2 times 20 = 400 ) years - New ( x ) trailer homes: ( x times 2 = 2x ) years [ text{Total age (now)} = 400 + 2x ]5. The average age now: [ frac{400 + 2x}{20 + x} = 14 ]We need to solve this equation to find ( x ). Let's do this step by step in Python.```pythonfrom sympy import symbols, Eq, solve # Define the variable x = symbols('x') # Total age two years ago total_age_two_years_ago = 360 # Total age now total_age_now = total_age_two_years_ago + 2 * 20 + 2 * x # Adding the age from the last two years for old and new homes # Total number of homes now total_homes_now = 20 + x # Average age equation average_age_equation = Eq(total_age_now / total_homes_now, 14) # Solve for x new_homes_added = solve(average_age_equation, x)[0] print(new_homes_added.evalf()) # Evaluate to get a numerical result``````output10.0000000000000```The solution to the equation indicates that the number of new trailer homes added two years ago is:[boxed{10}]Thus, 10 new trailer homes were added to Elm Street two years ago.