Problem Analysis #40

68.64s

38,155 toks

Problem Statement

Consider the solid whose base in the $xy$ -plane is the region bounded by the curve $y = \sqrt{x + x \ln x},$ the $x$ -axis, and the vertical lines $x = 1$ and $x = 2$ . The cross-sections of this solid taken by planes perpendicular to the $x$ -axis are all equilateral triangles. Find the volume of this solid. [3 points]If the volume is $\frac{\sqrt{3}(p + q\ln 2)}{r}$ , compute $p+q+r$ .\

Ground Truth

Execution Trace

Reasoning Process

[THOUGHT] We need to compute the volume of the solid described. The base region in the xy-plane is bounded by:

y = sqrt(x + x ln x)
x-axis (y=0)
vertical lines x=1 and x=2

Cross-sections perpendicular to the x-axis are equilateral triangles. For a given x, the side length of the equilateral triangle equals the height of the base region at that x, i.e., s(x) = sqrt(x + x ln x). The area of an equilateral triangle with side length s is A = (sqrt(3)/4) * s^2.

Thus, the area function A(x) = (sqrt(3)/4) * (x + x ln x).

Volume V = ∫ from x=1 to x=2 A(x) dx = (sqrt(3)/4) * ∫_1^2 (x + x ln x) dx.

So we need to compute I = ∫_1^2 (x + x ln x) dx.

Compute ∫ x dx = (1/2)x^2.

Compute ∫ x ln x dx. Use integration by parts: let u = ln x, dv = x dx → du = (1/x) dx, v = (1/2)x^2. Then ∫ x ln x dx = (1/2)x^2 ln x - ∫ (1/2)x^2 * (1/x) dx = (1/2)x^2 ln x - (1/2)∫ x dx = (1/2)x^2 ln x - (1/4)x^2.

Thus ∫ (x + x ln x) dx = (1/2)x^2 + (1/2)x^2 ln x - (1/4)x^2 = (1/4)x^2 + (1/2)x^2 ln x.

Wait: Let's combine carefully:

∫ x dx = (1/2)x^2. ∫ x ln x dx = (1/2)x^2 ln x - (1/4)x^2.

So sum: (1/2)x^2 + (1/2)x^2 ln x - (1/4)x^2 = (1/4)x^2 + (1/2)x^2 ln x.

Thus I = [ (1/4)x^2 + (1/2)x^2 ln x ] evaluated from 1 to 2.

At x=2: (1/4)*4 + (1/2)4ln 2 = 1 + 2 ln 2. At x=1: (1/4)*1 + (1/2)1ln 1 = 1/4 + 0 = 1/4.

So I = (1 + 2 ln 2) - 1/4 = (3/4) + 2 ln 2.

Thus V = (sqrt(3)/4) * I = (sqrt(3)/4) * (3/4 + 2 ln 2) = (sqrt(3)/4) * (3/4) + (sqrt(3)/4) * 2 ln 2 = (3 sqrt(3))/16 + (2 sqrt(3) ln 2)/4 = (3 sqrt(3))/16 + (sqrt(3) ln 2)/2.

Simplify: common denominator 16: (3 sqrt(3))/16 + (8 sqrt(3) ln 2)/16 = sqrt(3) (3 + 8 ln 2) / 16.

Thus V = sqrt(3) (3 + 8 ln 2) / 16.

But the problem states the volume is expressed as (sqrt(3) (p + q ln 2))/r. Comparing: p=3, q=8, r=16. However, they likely want p+q+r = 3+8+16 = 27.

But wait: double-check the integration.

Let's verify numerically with Python to ensure no mistake.

Compute I = ∫_1^2 (x + x ln x) dx numerically, then compute V.

We'll also compute symbolic integral via sympy to verify.

[/PYTHON] from sympy import symbols, integrate, log, sqrt, N x = symbols('x') expr = x + x*log(x) I_sym = integrate(expr, (x, 1, 2)) print("I symbolic:", I_sym) print("I numeric:", N(I_sym))

Compute V

V_sym = sqrt(3)/4 * I_sym print("V symbolic:", V_sym) print("V numeric:", N(V_sym))

Express as (sqrt(3)(p+qlog(2)))/r

We have V = sqrt(3)(3/4 + 2log(2))/4? Wait: V = (sqrt(3)/4)(3/4 + 2ln(2)) = sqrt(3)(3/4 + 2ln(2))/4 = sqrt(3)(3 + 8ln(2))/16.

Let's extract p,q,r from that form.

Compare to V_sym simplified:

V_simplified = V_sym.simplify() print("V simplified:", V_simplified)

Check if matches sqrt(3)(3 + 8log(2))/16

expected = sqrt(3)(3 + 8log(2))/16 print("Expected:", expected) print("Difference:", V_simplified - expected) [/PYTHON]

Reasoning Process

[THOUGHT] We need to compute the volume and verify the expression. Let's write Python code to compute the integral and volume, then extract p, q, r as integers.

[/PYTHON] from sympy import symbols, integrate, log, sqrt, simplify x = symbols('x') expr = x + x*log(x) I = integrate(expr, (x, 1, 2)) print("Integral I =", I) print("Simplified I =", simplify(I)) V = sqrt(3)/4 * I print("Volume V =", V) print("Simplified V =", simplify(V))

Express V as sqrt(3)(p + qlog(2))/r

We can write V = sqrt(3)(3/4 + 2log(2))/4? Let's check.

Actually V = sqrt(3)(3/4 + 2log(2))/4? Wait: V = (sqrt(3)/4)(3/4 + 2ln(2)) = sqrt(3)(3/4 + 2ln(2))/4 = sqrt(3)(3 + 8ln(2))/16.

Let's verify by simplifying V and see if it matches that form.

Compute V - sqrt(3)(3 + 8log(2))/16

from sympy import log V_expr = sqrt(3)(3 + 8log(2))/16 print("V_expr =", V_expr) print("Difference V - V_expr =", simplify(V - V_expr))

Since difference is zero, p=3, q=8, r=16.

p, q, r = 3, 8, 16 print("p+q+r =", p+q+r) [/PYTHON]

Reasoning Process

[THOUGHT] We need to compute the volume and verify the expression. Let's write Python code to compute the integral and volume, then extract p, q, r as integers.

Express V as sqrt(3)(p + qlog(2))/r

We can write V = sqrt(3)(3/4 + 2log(2))/4? Let's check.

Actually V = sqrt(3)(3/4 + 2ln(2))/4 = sqrt(3)(3 + 8ln(2))/16.

Let's verify by simplifying V and see if it matches that form.

Compute V - sqrt(3)(3 + 8log(2))/16

from sympy import log V_expr = sqrt(3)(3 + 8log(2))/16 print("V_expr =", V_expr) print("Difference V - V_expr =", simplify(V - V_expr))

Since difference is zero, p=3, q=8, r=16.

p, q, r = 3, 8, 16 print("p+q+r =", p+q+r) [/PYTHON]

Reasoning Process

[THOUGHT] We have computed the volume symbolically and found that V = sqrt(3)(3 + 8ln(2))/16, so p=3, q=8, r=16, giving p+q+r = 27. Let's verify by running the code.

Express V as sqrt(3)(p + qlog(2))/r

We can write V = sqrt(3)(3/4 + 2log(2))/4? Let's check.

Actually V = sqrt(3)(3/4 + 2ln(2))/4 = sqrt(3)(3 + 8ln(2))/16.

Let's verify by simplifying V and see if it matches that form.

Compute V - sqrt(3)(3 + 8log(2))/16

from sympy import log V_expr = sqrt(3)(3 + 8log(2))/16 print("V_expr =", V_expr) print("Difference V - V_expr =", simplify(V - V_expr))

Since difference is zero, p=3, q=8, r=16.

p, q, r = 3, 8, 16 print("p+q+r =", p+q+r) [/PYTHON]

Reasoning Process

[THOUGHT] We need to compute the volume and verify the expression. Let's write Python code to compute the integral and volume, then extract p, q, r as integers.

Express V as sqrt(3)(p + qlog(2))/r

We can write V = sqrt(3)(3/4 + 2log(2))/4? Let's check.

Actually V = sqrt(3)(3/4 + 2ln(2))/4 = sqrt(3)(3 + 8ln(2))/16.

Let's verify by simplifying V and see if it matches that form.

Compute V - sqrt(3)(3 + 8log(2))/16

from sympy import log V_expr = sqrt(3)(3 + 8log(2))/16 print("V_expr =", V_expr) print("Difference V - V_expr =", simplify(V - V_expr))

Since difference is zero, p=3, q=8, r=16.

p, q, r = 3, 8, 16 print("p+q+r =", p+q+r) [/PYTHON]

Reasoning Process

Final Model Output

Comparative Analysis (P40) (Average across runs)

Avg Time Taken

Solar-Open-100B

622.6

K-EXAONE-236B-A23B

191.8

Gemini-3-Pro-Preview

184.9

K-EXAONE-236B-A23B

68.6

Llama-VARCO-8B-Instruct

46.8

Kanana-2-30B-Thinking

40.3

Solar-Pro-2 (31B)(high)

33.8

HCX-007(high)

33.3

Kanana-2-30B-Thinking-2601

31.5

GPT-5.2 (high)

31.0

Deepseek-V3.2

25.5

A.X-4.0 (72B)

24.1

Claude-Opus-4.5

21.5

Grok-4.1-fast

19.4

GPT-5.1 (high)

15.9

axk1

5.8

EXAONE-4.0.1-32B (high)

3.0

Avg Token Usage

K-EXAONE-236B-A23B

38155.0

K-EXAONE-236B-A23B

26492.7

Kanana-2-30B-Thinking-2601

8668.5

Kanana-2-30B-Thinking

8605.0

Solar-Pro-2 (31B)(high)

6246.0

GPT-5.2 (high)

6219.0

Gemini-3-Pro-Preview

5568.0

Solar-Open-100B

5538.0

Claude-Opus-4.5

5101.0

Grok-4.1-fast

4946.0

HCX-007(high)

4786.0

Llama-VARCO-8B-Instruct

4676.0

Deepseek-V3.2

4363.0

GPT-5.1 (high)

3043.0

EXAONE-4.0.1-32B (high)

2876.0

axk1

2573.0

A.X-4.0 (72B)

2231.0