Sector Area and Function Integration

00:00 12 marks

The provided diagram illustrates a sector of a circle with centre O and radius r. The angle AOB is θ radians, and the perimeter of the sector is 24 cm.

(a) Show that the area of the sector, A, is given by \(A = 12r – r^2\). [4 marks]
b) Find the maximum possible area of the sector. [4 marks]

Consider the function \(f(x) = 3\sin\left(\frac{\pi x}{2}\right) + 1\) for \(0 \le x \le 4\).

Hint Hint

Hint

For part (a), use the perimeter formula \(2r + r\theta = 24\) to express \(\theta\) in terms of r and substitute into the sector area formula. In part (b), differentiate the expression obtained in part (a) and solve for r to maximize the area. In part (c), set up and evaluate the definite integral of \(3\sin(\pi x/2) + 1\) from 0 to 4.

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Answer Answer

Complete Answer

Part (a): \(A = 12r – r^2\). Part (b): Maximum area occurs when \(r = 6\) cm, giving \(A = 36\) cm². Part (c): The area under the curve is 4.

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