Consider the solid whose base in the -plane is the region bounded by the curvethe -axis, and the vertical lines and . The cross-sections of this solid taken by planes perpendicular to the -axis are all equilateral triangles. Find the volume of this solid. [3 points]If the volume is , compute .\
[THOUGHT] I will compute the integral for the volume using Python and simplify it into the required form. [/THOUGHT] [PYTHON] import sympy as sp
x = sp.symbols('x') expr = x + x*sp.log(x) V = sp.sqrt(3)/4 * sp.integrate(expr, (x,1,2)) V_simplified = sp.simplify(V) print("Volume:", V_simplified)
Express V as sqrt(3)(p + qln(2))/r
p, q, r = sp.symbols('p q r', integer=True)
Solve for (p + q ln 2)/r = the coefficient of sqrt(3)
coeff = sp.simplify(V_simplified / sp.sqrt(3)) print("Coefficient without sqrt(3):", coeff) coeff_rational = sp.together(coeff) print("Together:", coeff_rational)
Extract p, q, r manually from coeff
num, den = sp.fraction(coeff_rational) print("Numerator:", num) print("Denominator:", den)
Numerator should be of form p + q*ln(2)
p_extracted = sp.simplify(num.subs(sp.log(2), 0)) q_extracted = sp.simplify(sp.expand((num - p_extracted)/sp.log(2))) r_extracted = den
print("p:", p_extracted) print("q:", q_extracted) print("r:", r_extracted) print("p+q+r:", p_extracted + q_extracted + r_extracted) [/PYTHON]
[THOUGHT] From the Python result, the volume is (\sqrt{3}(3 + \ln 256)/16). Since (\ln 256 = \ln(2^8) = 8\ln 2), this becomes (\sqrt{3}(3 + 8\ln 2)/16). Thus (p=3), (q=8), (r=16), and (p+q+r=27). \boxed{27}