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Exam-Style Question on Volume of Revolution

A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 637. This question is similar to one that appeared on an IB AA Higher paper in 2022. The use of a calculator is allowed.

Consider the function \(f(x) = \sec^{-1}(x) \text{ where } 1 \le x \le \frac{\pi}{2} \)

(a) Sketch the curve \(y=f(x)\) clearly indicating the coordinates of the endpoints.

(b) State the domain and range of \( f(x) \).

The curve \(y=f(x) \) is rotated \(2\pi\) about the y-axis to form a solid of revolution that is used to model a flower vase.

(c) Find an expression for the volume, \(Vm^2\), of water in the vase when it is filled to a height of \(h\) metres.

(d) Hence determine the maximum volume of the vase.

At \(t = 0\) the vase is empty. Water is then added to the vase at a constant rate of \(0.1m^3s^{-1} \)

(e) Find the time it takes to fill the vase to Its maximum volume.

(f) Find the rate of change of the height of the water when the vase is filled to half its maximum height.

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