Related rates
Problem 3.97 · medium
A 5 ft tall person walks away from a 20 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: 20/(x + s) = 5/s, so s = 5x/15.
- \[ \frac{d}{d x} \frac{x}{3} = \frac{1}{3} \]ds/dt = (ds/dx)(dx/dt).✓ Proved
- \[ \frac{2}{3} \]Multiply by dx/dt = 2.✓ Proved
Answer \( \frac{2}{3} \ \text{ft/s} \approx 0.6667\ \text{ft/s} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 |
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 2026-09-26 with SymPy 1.14.0.