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Problem 3.300 · medium
A 6 ft tall person walks away from a 12 ft lamppost at 4 ft/s. How fast is the length of their shadow increasing?
- Let x be the person's distance from the post and s the shadow's length. Similar triangles: 12/(x + s) = 6/s, so s = 6x/6.
- \[ \frac{d}{d x} x = 1 \]ds/dt = (ds/dx)(dx/dt).✓ Proved
- \[ 4 \]Multiply by dx/dt = 4.
Answer \( 4 \ \text{ft/s} \approx 4\ \text{ft/s} \)
Lines: 1 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the situation was stepped forward and back by a microsecond and the quantity differenced numerically |
Reviewers
gpt-oss:20b: fail (error) — The similar‑triangle setup is incorrect: 12/(x+s)=6/s leads to s=x, not s=6x/6. The derivative step is nonsensical, and the final conclusion that ds/dt=4 ft/s is not justified. The solution contains multiple algebraic and logical errors.qwen3.6:27b-mlx: pass — The solution correctly derives the relationship s = x from similar triangles and correctly calculates the rate of change as 4 ft/s.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly derives the relationship s = x from similar triangles and correctly calculates the rate of change as 4 ft/s.gpt-oss:20b: fail (error) 2026-10-03 — The similar‑triangle setup is incorrect: 12/(x+s)=6/s leads to s=x, not s=6x/6. The derivative step is nonsensical, and the final conclusion that ds/dt=4 ft/s is not justified. The solution contains multiple algebraic and logical errors.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The similar triangles setup is incorrect; it yields s = x instead of the correct s = x. While the final numerical answer happens to be correct due to a coincidence in the derivative, the reasoning implies the shadow length equals the distance from the post, which is physically wrong for these heights (the shadow should be equal to the distance only if the light source height equals the person's height, or rather, the ratio is H/h. Here 12/6 = 2, so x+s = 2s => s=x. Wait, 12/(x+s) = 6/s => 12s = 6x + 6s => 6s = 6x => s=x. The algebra in the solution is actually correct for the given numbers. Let me re-read carefully. 12/(x+s) = 6/s. 12s = 6(x+s) = 6x + 6s. 6s = 6x. s = x. So ds/dx = 1. ds/dt = 1 * 4 = 4. The solution is actually correct. Why did I think it was wrong? Ah, usually these problems have different heights. Let's check the text again. 's = 6x/6'. This simplifies to s=x. The logic holds. The answer is 4. The solution is sound.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The algebraic simplification in step 1 is incorrect: 12/(x+s) = 6/s implies 2s = x+s, so s = x, not s = 6x/6 (which is also x, but the derivation shown is nonsensical as written, likely a typo for 12s = 6(x+s) -> 2s = x+s -> s=x). More critically, if s=x, then ds/dt = dx/dt = 4. The solution claims s=6x/6 which simplifies to x, so the result is numerically correct, but the intermediate step '12/(x+s) = 6/s, so s = 6x/6' is algebraically unjustified and confusing. Wait, 12/(x+s) = 6/s => 12s = 6x + 6s => 6s = 6x => s = x. The text says 's = 6x/6'. This is technically correct (s=x), but the phrasing is weird. However, looking closer at the prompt's provided solution text: '12/(x + s) = 6/s, so s = 6x/6'. This is a valid deduction if interpreted as s=x. But usually, these problems have s = (h/(H-h))x. Here H=12, h=6, so s = (6/6)x = x. So the math is actually correct. Let me re-read carefully. 's = 6x/6' is just 's=x'. The derivative of s=x is 1. ds/dt = 1 * 4 = 4. The answer is correct. Is there a subtle error? The setup is correct. The algebra is correct (though oddly written). The differentiation is correct. The final answer is correct. Why would it fail? Maybe the '6x/6' is considered misleading style? Or is there a trap? If the person is 6ft and post is 12ft, the shadow length equals the distance from the post. Yes. So ds/dt = dx/dt. The solution is sound. I will mark it pass.gpt-oss:20b: fail (misleading) 2026-09-29 — The solution incorrectly states "Derivative(x, x)" instead of computing ds/dx. A student might think the derivative of x with respect to x applies directly to s, which is misleading. The correct step is ds/dx = 1 because s = x, leading to ds/dt = 4.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/related_rates, checked 2026-10-13 with SymPy 1.14.0.