Problem Analysis #41

[THOUGHT] We are given a parametric curve and need to find the slope of the tangent at the intersection point with the line y = 3x - 5e. The slope is given in the form (aπ - b)/(π + c), and we need to compute a + b + c. First, find the intersection point by equating y(t) and 3x(t) - 5e, which yields t = 1/4. Then, compute the derivatives dx/dt and dy/dt at t = 1/4, and use dy/dx = (dy/dt) / (dx/dt) to find the slope. Using sympy, we will compute the expressions and evaluate at t=1/4 to verify the form. [/THOUGHT] [PYTHON] from sympy import * t = symbols('t') x = exp(4t) * (1 + sin(pit)**2) y = exp(4t) * (1 - 3cos(pi*t)**2)

dxdt = x.diff(t) dydt = y.diff(t)

slope_expr = dydt / dxdt slope_val = slope_expr.subs(t, 1/4)

print(str(slope_val)) [/PYTHON]

[THOUGHT] The parametric equations are given, and the intersection point with the line (y = 3x - 5e) occurs at (t = 1/4). The slope of the tangent is calculated as (\frac{dy/dt}{dx/dt}) evaluated at (t = 1/4), resulting in the expression (\frac{8.15484548537714\pi - 5.43656365691809}{2.71828182845905\pi + 16.3096909707543}). By recognizing that (8.15484548537714 \approx 3e) and (5.43656365691809 \approx 2e), and similarly (2.71828182845905 \approx e) and (16.3096909707543 \approx 6e), the expression simplifies to (\frac{e(3\pi - 2)}{e(\pi + 6)} = \frac{3\pi - 2}{\pi + 6}). Thus, (a = 3), (b = 2), (c = 6), and (a + b + c = 11). \boxed{11}