Consider the functionand let be a positive real number. For , let be a point on the curve . Let denote the value of such that the distance between the foot of the perpendicular from to the -axis and the point where the tangent to the curve at meets the -axis is equal to . Evaluate[4 points]\subsection*{Numerical answer}If the value is , compute .\
157 + 12 = 169
[THOUGHT] I need to evaluate the integral ∫_{1/2}^{27/4} g(t) dt for the given function and definition of g(t). After deriving the relationship t = s^3/(1+s), I substitute to express the integral in terms of s from s=1 to s=3. The integrand simplifies to (2s^4 + 3s^3)/(1+s)^2, which I transform via u = s+1, leading to a rational function in u from u=2 to u=4. I then use Python to verify the definite integral calculation symbolically. [/THOUGHT] [PYTHON] from sympy import symbols, integrate, ln u = symbols('u') expr = 2u**2 - 5u + 3 + 1/u - 1/u**2 integral_expr = integrate(expr, u) u_sym = symbols('u') integral_at_4 = integral_expr.subs(u_sym, 4) integral_at_2 = integral_expr.subs(u_sym, 2) result = integral_at_4 - integral_at_2 print(result) [/PYTHON]
[ \boxed{169} ]