answer:To solve this problem, we need to use the properties of even functions. An even function, by definition, satisfies the property ( f(x) = f(-x) ) for all ( x ) in its domain. This means that the value of the function at a positive ( x ) is the same as the value of the function at the corresponding negative ( x ).Given that ( f(x) ) is an even function and ( f(4) = 3 ), we can use the property of even functions to determine ( f(-4) ).Let's break down the steps:1. Use the property of even functions to determine ( f(-4) ).2. Compute the sum ( f(4) + f(-4) ).Let's implement this in Python using SymPy:```pythonimport sympy as sp# Given valuesf_4 = 3# Since f(x) is an even function, f(-4) = f(4)f_neg4 = f_4# Calculate the sumresult = f_4 + f_neg4print(result)``````output6```The value of ( f(4) + f(-4) ) is (boxed{6}).

answer:To solve the problem of finding the greatest common divisor (GCD) of 228 and 1995, we can use Python and the `sympy` library. The `sympy` library provides a convenient function `gcd` for calculating the greatest common divisor of two numbers.Here's a step-by-step approach to solving the problem:1. Import the `sympy` library.2. Use the `gcd` function to find the greatest common divisor of 228 and 1995.3. Print the result.Let's implement this in Python:```pythonimport sympy as sp# Define the two numbersnum1 = 228num2 = 1995# Calculate the greatest common divisor (GCD)gcd_result = sp.gcd(num1, num2)# Print the resultprint(gcd_result)``````output57```The greatest common divisor (GCD) of 228 and 1995 is (boxed{57}).The greatest common divisor (GCD) of 228 and 1995 is (boxed{57}).

answer:To determine the total height of all four buildings, we need to calculate the height of each building based on the given proportions and then sum them up.1. The height of the tallest building is given as (100) feet.2. The second tallest is half the height of the tallest building.3. The third tallest is half the height of the second tallest.4. The fourth tallest is one-fifth the height of the third tallest.We can write these as:- Height of 1st building = (100) feet- Height of 2nd building = (100 times frac{1}{2}) feet- Height of 3rd building = ((100 times frac{1}{2}) times frac{1}{2}) feet- Height of 4th building = (((100 times frac{1}{2}) times frac{1}{2}) times frac{1}{5}) feetNext, let's calculate these heights and find their sum.```python# Heights calculationheight_1st = 100height_2nd = height_1st * 0.5height_3rd = height_2nd * 0.5height_4th = height_3rd * 0.2# Total height of all buildingstotal_height = height_1st + height_2nd + height_3rd + height_4thprint(total_height)``````output180.0```The total height of all four buildings put together is (boxed{180}) feet.

answer:To determine who will receive the sixth candy wrapper, let's analyze the pattern in which Pete and Vasey are receiving the wrappers. From the problem statement:- The first three wrappers were obtained by Pete.- The next two wrappers were obtained by Vasey.- We need to determine who will receive the sixth wrapper.Let's break down the problem:1. Pattern Understanding: - Pete receives the first three wrappers. - Vasey receives the next two wrappers. - This pattern then repeats.2. Pattern Cycle: - Every complete cycle consists of 5 wrappers (3 for Pete and 2 for Vasey).3. Determine the Recipient: - To determine who gets the sixth wrapper, we find the position of the sixth wrapper within its cycle.The cycle length is 5 (3 for Pete + 2 for Vasey). [text{Index of the sixth wrapper within its cycle} = 6 mod 5 = 1]Since the remainder is 1, it falls within the first three positions of the cycle, which are assigned to Pete. Therefore, Pete will receive the sixth candy wrapper.Let's confirm this reasoning with a Python code to validate the result.```pythondef find_recipient(wrapper_number): cycle_length = 5 pete_wrappers = 3 vasey_wrappers = 2 # Find the position within the cycle position_in_cycle = wrapper_number % cycle_length # Determine the recipient based on the position in the cycle if 1 <= position_in_cycle <= pete_wrappers: return "Pete" elif pete_wrappers < position_in_cycle <= pete_wrappers + vasey_wrappers: return "Vasey" else: return "Unknown"# Determine the recipient of the sixth wrappersixth_wrapper_recipient = find_recipient(6)print(sixth_wrapper_recipient)``````outputPete```The Python code confirms that the recipient of the sixth candy wrapper is (boxed{text{Pete}}).

answer:Given the problem of dividing a cube with an edge length of ( 3 , text{cm} ) into ( n ) smaller cubes with whole centimeter edge lengths, we need to determine the value of ( n ).Let's lay out the reasoning step by step:1. The volume of the original cube is ( 3^3 = 27 , text{cm}^3 ).2. We need to decompose this volume into smaller cubes where each of the smaller cubes has volumes that can only be ( 1 , text{cm}^3 ), ( 8 , text{cm}^3 ), or ( 27 , text{cm}^3 ), corresponding to edge lengths of 1 cm, 2 cm, and 3 cm respectively. Let's check for possible combinations (keeping in mind integer solutions to volume equations):If we denote:- ( x ) as the number of ( 1 , text{cm}^3 ) cubes,- ( y ) as the number of ( 8 , text{cm}^3 ) cubes, and- ( z ) as the number of ( 27 , text{cm}^3 ) cubes,Then we have the following system of equations:[ x + y + z = n ][1x + 8y + 27z = 27]We need solutions where ( x, y, ) and ( z ) are non-negative integers. Let's solve for the feasible values of ( x, y, ) and ( z ) that satisfy the volume equation.```pythonimport itertools# The total volume of the original cubetotal_volume = 27# We test for values of x, y, z such that:# x * 1 + y * 8 + z * 27 = 27# Iterate through potential values of x, y, zsolutions = []for x in range(total_volume + 1): for y in range(total_volume // 8 + 1): for z in range(total_volume // 27 + 1): if x * 1 + y * 8 + z * 27 == total_volume: solutions.append((x, y, z, x + y + z))print(solutions)``````output[(0, 0, 1, 1), (3, 3, 0, 6), (11, 2, 0, 13), (19, 1, 0, 20), (27, 0, 0, 27)]```The feasible solutions for ((x, y, z)) that satisfy (x cdot 1 + y cdot 8 + z cdot 27 = 27) along with (x + y + z = n) are:1. ((0, 0, 1)) yielding (n = 1)2. ((3, 3, 0)) yielding (n = 6)3. ((11, 2, 0)) yielding (n = 13)4. ((19, 1, 0)) yielding (n = 20)5. ((27, 0, 0)) yielding (n = 27)Among the provided options ((A) 4, (B) 8, (C) 12, (D) 16, (E) 20), the only value that matches is (n = 20).So, the final answer is (boxed{20}).

answer:To solve the problem, we need to follow these steps:1. Determine the Parametric Equation of Line (l): The line (l) passes through point (P(1,1)) with an inclination angle (alpha = frac{pi}{6}). 2. Find the Intersection Points (A) and (B) of Line (l) with Circle (C): Substitute the parametric equations of the circle into the equation of the line to find the points of intersection (A) and (B).3. Calculate the Distances from Point (P) to Points (A) and (B): Use the distance formula to determine the distances from (P) to (A) and from (P) to (B).4. Find the Product of the Distances: Multiply the distances to get the final result.Let's implement these steps using Python and SymPy. Step 1: Determine the Parametric Equation of Line (l)Since the line (l) has an inclination angle (alpha = frac{pi}{6}), the slope of the line is (tanleft(frac{pi}{6}right) = frac{1}{sqrt{3}}). Therefore, the equation of the line in slope-intercept form is:[y - 1 = frac{1}{sqrt{3}}(x - 1)] Step 2: Find the Intersection Points (A) and (B)Substitute (x = 2costheta) and (y = 2sintheta) into the equation of the line to find the intersection points. Step 3: Calculate the Distances from Point (P) to Points (A) and (B)Use the distance formula to find the distances. Step 4: Find the Product of the DistancesMultiply the distances of points (A) and (B) from (P).Let's implement the steps in Python using SymPy.```pythonimport sympy as sp# Define symbolstheta = sp.symbols('theta')x, y = sp.symbols('x y')# Parametric equations of the circlex_circle = 2 * sp.cos(theta)y_circle = 2 * sp.sin(theta)# Parametric equation of the linealpha = sp.pi / 6slope = sp.tan(alpha)P = (1, 1)x_line = sp.Symbol('x')y_line = P[1] + slope * (x_line - P[0])# Solve for the intersection pointsintersection_eq = sp.Eq(y_circle, y_line)theta_solutions = sp.solve(intersection_eq, theta)# Calculate the coordinates of the intersection points A and Bpoints = [(2 * sp.cos(sol), 2 * sp.sin(sol)) for sol in theta_solutions]# Calculate the distances from P to A and Bdistances = [sp.sqrt((P[0] - point[0])2 + (P[1] - point[1])2) for point in points]# Calculate the product of the distancesproduct_of_distances = distances[0] * distances[1]# Simplify the resultproduct_of_distances = sp.simplify(product_of_distances)# Print the resultprint(product_of_distances)``````outputsqrt(3*(x - 1)2 + (sqrt(36 - (sqrt(3)*x - sqrt(3) + 3)2) - 3)2)*sqrt(3*(x - 1)2 + (sqrt(36 - (sqrt(3)*x - sqrt(3) + 3)2) + 3)2)/9```It seems that the output is quite complex. Let's break down the steps again to ensure we simplify the operations and get a clearer result. We need to:1. Find the intersection points (A) and (B) by solving the parametric equations.2. Compute the distances from (P) to (A) and (B).3. Compute their product.Given the complexity of solving it all at once, let's take a more systematic step-by-step approach:1. Write the line equation in parametric form: [ begin{cases} x = t + 1 y = frac{1}{sqrt{3}}t + 1 end{cases} ]2. Substitute the line's parametric equations into the circle's equation to find (t):[ ( t + 1 )^2 + (frac{1}{sqrt{3}}t + 1)^2 = 4 ]3. Solve for (t), which gives us the intersection points.4. Use the distance formula to calculate the distances from (P) to (A) and (B).5. Multiply these distances to get the final product.Let's adjust the Python code to handle these steps more clearly:```pythonimport sympy as sp# Define symbolst = sp.symbols('t')x, y = sp.symbols('x y')# Parametric equations of the linex_line = t + 1y_line = (1/sp.sqrt(3))*t + 1# Circle equation: x^2 + y^2 = 4circle_eq = sp.Eq(x_line2 + y_line2, 4)# Solve for t to find the intersection pointst_solutions = sp.solve(circle_eq, t)# Calculate the coordinates of the intersection points A and Bpoints = [(t_sol + 1, (1/sp.sqrt(3))*t_sol + 1) for t_sol in t_solutions]# Point PP = (1, 1)# Calculate the distances from P to A and Bdistances = [sp.sqrt((P[0] - point[0])2 + (P[1] - point[1])2) for point in points]# Calculate the product of the distancesproduct_of_distances = distances[0] * distances[1]# Simplify the resultproduct_of_distances = sp.simplify(product_of_distances)# Print the resultprint(product_of_distances)``````output2```The product of the distances from point (P) to points (A) and (B) is (boxed{2}).To summarize, we found the parametric equation of the line (l), determined the intersection points (A) and (B) with the circle (C), calculated the distances from (P) to those points, and computed the product of those distances, which turned out to be 2.