Mathematics

Mathematics

Problem

A fish population in a lake obeys dydt=0.02 y(50−y)\dfrac{dy}{dt}=0.02\,y(50-y). Which initial population listed below would result in the greatest initial rate of increase? a) y(0)=50y(0)=50 b) y(0)=25y(0)=25 c) y(0)=30y(0)=30 d) y(0)=12.5y(0)=12.5
Intuitive explanation

This is a logistic growth problem where the rate of change depends on both the current population yy and how much "room" remains (the factor 50−y50-y). The key insight is that the initial rate of increase is maximized when the product y(50−y)y(50-y) is largest—not when yy is largest. Since this product is a downward-opening parabola, it peaks at the midpoint between 0 and 50, where the population has the most balanced "push" from both factors.

Intuitive explanation

This is a logistic growth problem where the rate of change depends on both the current population yy and how much "room" remains (the factor 50−y50-y). The key insight is that the initial rate of increase is maximized when the product y(50−y)y(50-y) is largest—not when yy is largest. Since this product is a downward-opening parabola, it peaks at the midpoint between 0 and 50, where the population has the most balanced "push" from both factors.

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