When more fertilizer stops helping, a relationship that bends

Question

A farm fit a straight line predicting crop yield from fertilizer applied, using amounts from $0$ to $50$ kg per acre. Over that range more fertilizer meant more yield, and the line fit well. An advisor now uses it to predict yield at $300$ kg per acre.

Explain why the prediction is likely to be badly wrong, referring to how the real relationship behaves.

Accepted Answer

The line predicts yield to keep climbing at the same rate all the way to 300 kg, but the real relationship almost certainly bends, so the forecast is not to be trusted. Diminishing returns, then harm. In the tested range (0 to 50 kg) each added kilogram helped, so a line fit fine. But fertilizer does not help without limit. Plants can only use so much; beyond a point extra fertilizer adds little, and past that it can burn roots and poison the soil, actually reducing yield. The true curve rises, levels off, and eventually turns down. A straight line captures none of this and just keeps rising.…

When more fertilizer stops helping, a relationship that bends

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