A rectangle has a fixed perimeter of $40\text{ cm}$. Its width $x$ increases at a constant rate

Question

A rectangle has a fixed perimeter of $40\text{ cm}$. Its width $x$ increases at a constant rate of $0.5\text{ cm s}^{-1}$, while its length adjusts to keep the perimeter fixed. Show that its area is $A=20x-x^2. $ Find $\frac{\mathrm{d}A}{\mathrm{d}t}$ when $x=8\text{ cm}$. Find $\frac{\mathrm{d}A}{\mathrm{d}t}$ when $x=10\text{ cm}$. A student says, “The answer to part (c) means the area stays constant after that instant.” Explain why this is incorrect. Support your explanation by finding the area $\tau$ seconds after the instant when $x=10$, for small positive $\tau$.

Accepted Answer

(b) $2\text{ cm}^2/\text{s}$ (c) $0\text{ cm}^2/\text{s}$; (d) Area $100-\tau^2/4\text{ cm}^2$.

Differentiation

Differentiation

Solution:

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