answer:To calculate the area enclosed by the cardioid (rho = 1 + cos varphi) for (0 leqslant varphi leqslant 2pi), we can use the formula for the area in polar coordinates:[ A = frac{1}{2} int_{0}^{2pi} rho^2 , dvarphi ]Given (rho = 1 + cos varphi), we need to square (rho) and integrate over the interval from (0) to (2pi).Let's break down the steps and write the Python code to compute this area using SymPy:1. Define the variable (varphi).2. Define (rho = 1 + cos varphi).3. Compute (rho^2).4. Set up and evaluate the integral (frac{1}{2} int_{0}^{2pi} rho^2 , dvarphi).Here's the Python code to solve the problem:```pythonimport sympy as sp# Define the variablephi = sp.Symbol('phi')# Define rhorho = 1 + sp.cos(phi)# Define rho squaredrho_squared = rho2# Set up the integral for the areaarea_integral = sp.integrate(rho_squared, (phi, 0, 2*sp.pi))# Calculate the areaarea = (1/2) * area_integral# Print the resultprint(area.evalf())``````output4.71238898038469```The area enclosed by the cardioid (rho = 1 + cos varphi) for (0 leq varphi leq 2pi) is (boxed{4.71238898038469}).This result is approximately equal to (frac{3pi}{2}), which is a well-known result for the area of a cardioid.

answer:To solve for ( 3-a ) given the equations ( 3 + a = 4 - b ) and ( 4 + b = 7 + a ), we can follow these steps:1. Solve the first equation ( 3 + a = 4 - b ) for one variable in terms of the other.2. Substitute the expression from step 1 into the second equation to solve for one variable.3. Use the value of the solved variable to find the other variable.4. Finally, calculate ( 3 - a ).Let's proceed with these steps using Python.```pythonfrom sympy import symbols, Eq, solve # Define the variables a, b = symbols('a b') # Define the equations eq1 = Eq(3 + a, 4 - b) eq2 = Eq(4 + b, 7 + a) # Solve eq1 for b b_expr = solve(eq1, b)[0] # Substitute b in eq2 and solve for a a_value = solve(eq2.subs(b, b_expr), a)[0] # Find the value of b using the value of a b_value = b_expr.subs(a, a_value) # Calculate 3 - a result = 3 - a_value print((a_value, b_value, result))``````output(-1, 2, 4)```The solutions for the variables are:[ a = -1 ][ b = 2 ]Now, using the value of ( a ):[ 3 - a = 3 - (-1) = 3 + 1 = 4 ]Thus, the final result is:[boxed{4}]

answer:To solve the inequality [5x - 1 < (x + 1)^2 < 7x - 3,]we need to split it into two separate inequalities and solve them step-by-step:1. (5x - 1 < (x + 1)^2)2. ((x + 1)^2 < 7x - 3)Let's solve these inequalities one by one using Python to ensure accuracy.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the inequalitiesineq1 = 5 * x - 1 < (x + 1)2ineq2 = (x + 1)2 < 7 * x - 3# Solve the inequalities using sympysol1 = sp.solve_univariate_inequality(ineq1, x, relational=False)sol2 = sp.solve_univariate_inequality(ineq2, x, relational=False)print((sol1, sol2))``````output(Union(Interval.open(-oo, 1), Interval.open(2, oo)), Interval.open(1, 4))```The solutions to the individual inequalities are:1. (5x - 1 < (x + 1)^2) results in (x in (-infty, 1) cup (2, infty)).2. ((x + 1)^2 < 7x - 3) results in (x in (1, 4)).To find the values of (x) that satisfy both inequalities simultaneously, we need to find the intersection of these two sets.The intersection of ( (-infty, 1) cup (2, infty) ) and ( (1, 4) ) is:[ (2, 4) ]Hence, the values of (x) that satisfy the original compound inequality are:[boxed{(2, 4)}]

answer:Let's start by solving the given equation step by step, and using Python code to ensure accuracy.The given equation is:[ 25^{log_{4} x} - 5^{log_{16} x^2 + 1} = log_{sqrt{3}} 9 sqrt{3} - 25^{log_{16} x} ]First, simplify the right-hand side of the equation.1. Simplify (log_{sqrt{3}} 9 sqrt{3}):[ log_{sqrt{3}} 9 sqrt{3} = log_{sqrt{3}} (9 cdot sqrt{3}) = log_{sqrt{3}} (3^2 cdot 3^{1/2}) = log_{sqrt{3}} 3^{(2 + 1/2)} = log_{sqrt{3}} 3^{5/2} ]Using the change of base formula, we get:[ log_{sqrt{3}} 3^{5/2} = frac{5/2 cdot log 3}{log sqrt{3}} = frac{5/2 cdot log 3}{frac{1}{2} cdot log 3} = 5 ]Now the equation becomes:[ 25^{log_{4} x} - 5^{log_{16} x^2 + 1} = 5 - 25^{log_{16} x} ]Next, let's solve step by step using Python to ensure accuracy.```pythonimport sympy as sp # Define the variable x = sp.symbols('x') # Define the left-hand side and right-hand side of the given equation lhs = 25(sp.log(x, 4)) - 5(sp.log(x2, 16) + 1) rhs = 5 - 25(sp.log(x, 16)) # Solve the equation solution = sp.solve(lhs - rhs, x) print(solution)``````output[4]```The solution to the equation ( 25^{log_{4} x} - 5^{log_{16} x^2 + 1} = log_{sqrt{3}} 9 sqrt{3} - 25^{log_{16} x} ) is:[ boxed{4} ]

answer:To solve this problem, we need to find the real part of the complex fraction (frac{1}{1 - z}) where (z) is a nonreal complex number such that (|z| = 1).Let's break down the problem into manageable steps:1. Parameterize the Complex Number (z): Since (|z| = 1), we can parameterize (z) as (z = e^{itheta}) where (theta) is a real number.2. Find the Expression for (frac{1}{1 - z}): Substitute (z = e^{itheta}) into the fraction (frac{1}{1 - z}).3. Simplify the Expression: Simplify (frac{1}{1 - e^{itheta}}) and find its real part.Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the variablestheta = sp.symbols('theta', real=True)z = sp.exp(sp.I * theta)# Define the expression 1 / (1 - z)expression = 1 / (1 - z)# Simplify the expressionexpression_simplified = sp.simplify(expression)# Find the real part of the expressionreal_part = sp.re(expression_simplified)print(real_part)``````output(1 - cos(theta))/((1 - cos(theta))2 + sin(theta)2)```The output ((1 - cos(theta))/((1 - cos(theta))^2 + sin(theta)^2)) represents the real part of the simplified expression (frac{1}{1 - e^{itheta}}).To simplify further, let's rewrite the denominator:[(1 - cos(theta))^2 + sin(theta)^2 ]Notice:[(1 - cos(theta))^2 = 1 - 2cos(theta) + cos^2(theta)][sin^2(theta) = 1 - cos^2(theta)]Thus:[(1 - cos(theta))^2 + sin^2(theta) = (1 - 2cos(theta) + cos^2(theta)) + (1 - cos^2(theta)) = 2 - 2cos(theta)]So the expression simplifies to:[frac{1 - cos(theta)}{2 - 2cos(theta)} = frac{1 - cos(theta)}{2(1 - cos(theta))} = frac{1}{2}]Therefore, the real part of (frac{1}{1 - z}) is (frac{1}{2}).Let's verify this simplification with Python and sympy:```pythonimport sympy as sp# Define the variablestheta = sp.symbols('theta', real=True)z = sp.exp(sp.I * theta)# Define the expression 1 / (1 - z)expression = 1 / (1 - z)# Simplify the expressionexpression_simplified = sp.simplify(expression)# Find the real part of the expressionreal_part = sp.re(expression_simplified)# Simplify the real part furtherreal_part_simplified = sp.simplify(real_part)print(real_part_simplified)``````output1/2```The output confirms that the real part of (frac{1}{1 - z}) is indeed (frac{1}{2}).Thus, the final answer is:[boxed{frac{1}{2}}]

answer:To solve the problem, we need to find the value of ((vec{a} + vec{b}) cdot vec{c}) given the conditions that (vec{a} perp vec{c}) and (vec{b} parallel vec{c}).Let's break down the problem into manageable steps:1. Understand the Conditions: - (vec{a} perp vec{c}) means the dot product (vec{a} cdot vec{c} = 0). - (vec{b} parallel vec{c}) means (vec{b} = k vec{c}) for some scalar (k). 2. Define the Vectors: - (vec{a} = (x, 1)) - (vec{b} = (1, y)) - (vec{c} = (3, -6))3. Use the Conditions to Find (x) and (y): - For (vec{a} perp vec{c}), we have ((x, 1) cdot (3, -6) = 0). - For (vec{b} parallel vec{c}), we have ((1, y) = k(3, -6)).4. Calculate ((vec{a} + vec{b}) cdot vec{c}): - First, find (vec{a} + vec{b}). - Then, compute the dot product with (vec{c}).Let's implement this in Python using sympy to find the value of ((vec{a} + vec{b}) cdot vec{c}).```pythonimport sympy as sp# Define variablesx, y, k = sp.symbols('x y k')# Define vectorsa = sp.Matrix([x, 1])b = sp.Matrix([1, y])c = sp.Matrix([3, -6])# Condition 1: a . c = 0condition1 = a.dot(c)# Condition 2: b = k * ccondition2 = b - k * c# Solve for x and ysol_x = sp.solve(condition1, x)[0]sol_y = sp.solve(condition2[1], y)[0]# Substitute x and y back into the vectorsa_sub = a.subs(x, sol_x)b_sub = b.subs(y, sol_y)# Find a + ba_plus_b = a_sub + b_sub# Compute the dot product (a + b) . cresult = a_plus_b.dot(c)print(result)``````output36*k + 3```From the Python code, we obtained the expression (36k + 3) for ((vec{a} + vec{b}) cdot vec{c}). Now, we need to find the appropriate value for (k).To do this, let's revisit the condition (vec{b} parallel vec{c}):- (vec{b} = (1, y))- (vec{c} = (3, -6))- Since (vec{b} parallel vec{c}), we have ((1, y) = k (3, -6))From this condition, we get two equations:[ 1 = 3k ][ y = -6k ]Solving the first equation for (k):[ k = frac{1}{3} ]Now, substituting (k = frac{1}{3}) into the expression (36k + 3):[36 left(frac{1}{3}right) + 3 = 12 + 3 = 15]So, the final value of ((vec{a} + vec{b}) cdot vec{c}) is (boxed{15}).