Mathematics

Mathematics

Problem

A rectangle’s area is kept constant for all time. At an instant when the rectangle’s length is increasing, determine whether the width is increasing, decreasing, or remaining constant, and state why in one sentence.
Intuitive explanation

The key is related rates and the constraint that area stays fixed. Since A=ℓ×wA = \ell \times w and AA is constant, when you differentiate you get dℓdt⋅w+ℓ⋅dwdt=0\frac{d\ell}{dt} \cdot w + \ell \cdot \frac{dw}{dt} = 0—this tells you the rates are linked. If length increases (dℓdt>0\frac{d\ell}{dt} > 0), the width must decrease to keep the product constant.

Intuitive explanation

The key is related rates and the constraint that area stays fixed. Since A=ℓ×wA = \ell \times w and AA is constant, when you differentiate you get dℓdt⋅w+ℓ⋅dwdt=0\frac{d\ell}{dt} \cdot w + \ell \cdot \frac{dw}{dt} = 0—this tells you the rates are linked. If length increases (dℓdt>0\frac{d\ell}{dt} > 0), the width must decrease to keep the product constant.

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