Problem Analysis #40

[THOUGHT] I need to find the volume of the solid with equilateral triangular cross-sections. The base is bounded by y = sqrt(x + x ln x), the x-axis, x=1, and x=2. The area of an equilateral triangle with side length s is (sqrt(3)/4)s². Here, s is the height function y = sqrt(x(1 + ln x)), so the area is (sqrt(3)/4) * [x(1 + ln x)]. The volume is the integral of this area from x=1 to x=2. I'll use sympy to compute the integral and express the volume in the form (sqrt(3)(p + q ln 2)) / r, then sum p, q, r. [/THOUGHT] [PYTHON] from sympy import symbols, log, integrate, sqrt x = symbols('x') integral_expr = x * (1 + log(x)) I = integrate(integral_expr, (x, 1, 2)) volume = (sqrt(3) / 4) * I print(volume) [/PYTHON]

[THOUGHT] The volume expression from the code is (\frac{\sqrt{3} ( \frac{3}{4} + 2 \ln 2 )}{4}). Rewriting this as (\frac{\sqrt{3} (p + q \ln 2)}{r}):

[ \frac{\sqrt{3} ( \frac{3}{4} + 2 \ln 2 )}{4} = \frac{\sqrt{3}}{4} \left( \frac{3}{4} + 2 \ln 2 \right) = \frac{\sqrt{3} (3 + 8 \ln 2)}{16} ]

Thus, (p = 3), (q = 8), (r = 16). Therefore, (p + q + r = 3 + 8 + 16 = 27). [/THOUGHT] \boxed{27}