Optimizing Rectangular Area with Perimeter Constraint

00:00 8 marks

A rectangle is to be constructed using a fixed length of fence with a total perimeter of 100 m.

Part a

4 minutes 3 marks

Question

Write an expression for the area, \(A(x)\), of the rectangle as a function of its length, \(x\).

Hint Hint

Hint

Use the perimeter condition to express the width in terms of the length, and then use the area formula.

Answer Answer
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Complete Answer

\(A(x) = x(50 – x)\)

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Part b

4 minutes 3 marks

Question

Find the value of \(x\) that maximizes the area by setting the derivative of \(A(x)\) equal to zero and solving for \(x\).

Hint Hint

Hint

Differentiate \(A(x) = 50x – x^2\) with respect to \(x\) and set the derivative equal to zero.

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Complete Answer

\(x = 25\)

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Part c

4 minutes 2 marks

Question

Interpret the result and state the dimensions of the rectangle which yield the maximum area.

Hint Hint

Hint

Use the value of \(x\) found in part (b) to determine the width \(y\) and state the dimensions.

Answer Answer
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Complete Answer

The rectangle with maximum area is a square with dimensions 25 m by 25 m.

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