Optimizing Dimensions Using First Principles – ++ Prep

Optimizing Dimensions Using First Principles

00:00 6 marks

A rectangle has a fixed perimeter of 40 meters. Let one side be \(x\) meters, so the adjacent side is \(20 – x\) meters.

(a) Express the area \(A(x)\) of the rectangle in terms of \(x\).

(b) Use the definition of the derivative (first principles) to derive an expression for \(A'(x)\).

(c) Determine the value of \(x\) that maximizes the area and state the corresponding dimensions of the rectangle.

Hint Hint

Hint

Write the area as \(A(x)=x(20-x)\). Then use the limit definition \(A'(x)=\lim_{h\to0}\frac{A(x+h)-A(x)}{h}\) to find the derivative.

Answer Answer
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Answer Answer

Complete Answer

The maximum area is achieved when \(x = 10\) m, so the rectangle is a square with dimensions 10 m by 10 m.

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