2024 DHS P1 Q10
2024 DHS P1 Q10
9 marks
Prelims
- A chef plans to create an ornament for his master dish. The ornament is made by rotating the shaded region as shown in the diagram completely about the \(y\)-axis. The region is bounded by the parabola \(y={{\left( x-3 \right)}^{2}}\), the curve \(C\) and the lines \(y=9+d\) and \(y=d\), where \(0 < d < 9\). The curve \(C\) is the reflection of the parabola along the line \(y=9\).
- Find the equation of the curve \(C\).[2]
- Find the volume of the ornament in terms of \({d}\).[5]
- By varying the value of \({d}\), find the maximum volume for the ornament correct to the nearest integer. State the value of \({d}\) corresponding to this maximum volume.[2]
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Answer:(b)(i) \({y = 18 - (x - 3)^2}\) (ii) \({8\pi \left[ 54 - (d)^{\frac{3}{2}} - (9 - d)^{\frac{3}{2}} \right]}\) (iii) \({877\text{ units}^3}\)
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