Related rates
Problem 3.339 · medium
A 6 ft tall person walks away from a 12 ft lamppost at 2 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
- \[ 2 \]Multiply by dx/dt = 2.✓ Proved
Answer \( 2 \ \text{ft/s} \approx 2\ \text{ft/s} \)
Lines: 2 proved, 1 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.
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 solution skips the crucial step of differentiating the relation s=x to obtain ds/dx=1, and the second line incorrectly presents a derivative of x with respect to x instead of ds/dx. This omission makes the reasoning incomplete and potentially misleading.qwen3.6:27b-mlx: fail (error) — The solution incorrectly simplifies s = 6x/6 to s = x, implying the shadow length equals the distance from the post, which is physically incorrect for a 6 ft person and 12 ft lamppost (s should be constant at 6 ft). This leads to the wrong derivative ds/dt = 2 instead of 0.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution incorrectly simplifies s = 6x/6 to s = x, implying the shadow length equals the distance from the post, which is physically incorrect for a 6 ft person and 12 ft lamppost (s should be constant at 6 ft). This leads to the wrong derivative ds/dt = 2 instead of 0.gpt-oss:20b: fail (error) 2026-10-03 — The solution skips the crucial step of differentiating the relation s=x to obtain ds/dx=1, and the second line incorrectly presents a derivative of x with respect to x instead of ds/dx. This omission makes the reasoning incomplete and potentially misleading.qwen3.6:27b-mlx: fail (error) 2026-10-03 — The similar triangles setup is incorrect; it should be 12/(x+s) = 6/s, which simplifies to s = x, not s = 6x/6 (which is just x, but the derivation shown implies a specific cancellation that hides the geometry). More importantly, the derivative step is logically flawed: it computes ds/dx = 1, then multiplies by dx/dt to get ds/dt = 2. While the final numerical answer happens to be correct for this specific setup (since s=x implies ds/dt = dx/dt), the intermediate step 's = 6x/6' is a tautology that doesn't show the work, and the explanation 'Derivative(x, x) = 1' is not a standard mathematical statement. However, the primary error is that the solution claims s = 6x/6 derived from 12/(x+s)=6/s. Let's check: 12s = 6(x+s) => 12s = 6x + 6s => 6s = 6x => s=x. So s=x is correct. The derivative ds/dx = 1 is correct. ds/dt = 1 * 2 = 2. The answer is actually correct. Why did I think it was wrong? Let me re-read carefully. Wait, the prompt says '12/(x + s) = 6/s'. This is the correct similar triangles ratio (Height of post / Total distance = Height of person / Shadow length). 12/(x+s) = 6/s 12s = 6x + 6s 6s = 6x s = x. So s = x. ds/dt = dx/dt = 2. The solution says 's = 6x/6'. This is equal to x. Then it says 'Derivative(x, x) = 1'. This likely means d/dx(x) = 1, so ds/dx = 1. Then '2 = 2' is ds/dt = (ds/dx)(dx/dt) = 1 * 2 = 2. The solution is actually mathematically sound, albeit poorly explained. The 'sentence' part is 'Let x be... Similar triangles: ... so s = 6x/6.' This is a correct derivation. The equations are marked unchecked but the logic holds. Is there a subtle error? Usually, these problems have s = (h/H - 1)x or similar. Here h=6, H=12. s = (6/12 / (1 - 6/12)) x ? No. 12/(x+s) = 6/s => 2/s = 1/(x+s) => 2x + 2s = s => 2x = -s? No. 12/(x+s) = 6/s => 12s = 6x + 6s => 6s = 6x => s=x. Yes, s=x is correct. So why would this fail? The prompt asks me to find what is wrong. Maybe the issue is 's = 6x/6'. It's weird notation but not wrong. Maybe the issue is the explanation of the derivative. 'Derivative(x, x) = 1' is not a standard sentence. It looks like code. But the instructions say: 'Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for.' Line 1 is a sentence. It sets up the model. The model is correct. Line 2 is an equation. Line 3 is an equation. If the solution is correct, I should pass. However, often in these adversarial reviews, 's = 6x/6' is considered 'style' or 'misleading' if it obscures the fact that s=x. But it's not false. Let's look closer at the similar triangles. Post height H = 12. Person height h = 6. Distance from post to person = x. Shadow length = s. Total distance from post to tip of shadow = x + s. Triangle 1: Height 12, Base x+s. Triangle 2: Height 6, Base s. Ratio: 12/(x+s) = 6/s. This is correct. Is it possible the problem implies the shadow length is measured from the person's feet? Yes, 's the shadow's length'. Is it possible the answer is different? ds/dt = 2. I will mark it as pass because the math is correct, even if the presentation is terse.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.