Problem Analysis #40

[THOUGHT] To find the volume of the solid, I need to integrate the area of the equilateral triangles along the x-axis from x=1 to x=2. The side length of each equilateral triangle is equal to the height of the curve y = sqrt(x + x ln x) at each x. The area of an equilateral triangle with side length s is (sqrt(3)/4 * s^2). So, the volume V is the integral from 1 to 2 of (sqrt(3)/4 * y^2) dx, where y = sqrt(x + x ln x). Simplifying, V = (sqrt(3)/4) * integral from 1 to 2 of (x + x ln x) dx. I will compute this integral using Python. [/THOUGHT] [PYTHON] import sympy as sp

x = sp.symbols('x') integrand = x + x * sp.ln(x) integral = sp.integrate(integrand, (x, 1, 2)) volume = (sp.sqrt(3)/4) * integral volume_simplified = sp.simplify(volume) print(volume_simplified) [/PYTHON]

[THOUGHT] The Python output simplifies to $\frac{\sqrt{3}(3 + \ln 256)}{16}$. Since $\ln 256 = 8 \ln 2$, the expression becomes $\frac{\sqrt{3}(3 + 8 \ln 2)}{16}$. Comparing with the given form $\frac{\sqrt{3}(p + q \ln 2)}{r}$, we have $p = 3$, $q = 8$, and $r = 16$. Thus, $p + q + r = 3 + 8 + 16 = 27$. [/THOUGHT]

$\boxed{27}$