A flat novelty plate for serving food on is made in the shape of the region enclosed by the curve \(

Question

A flat novelty plate for serving food on is made in the shape of the region enclosed by the curve $y={{x}^{2}}-6x+5$ and the line $2y=x-1$. Find the area of the plate.[4] A curved container has a flat circular top. The shape of the container is formed by rotating the part of the curve $x=\frac{\sqrt{y}}{a-{{y}^{2}}}$, where $a$ is a constant greater than $1$, between the points $(0,0)$ and $\left( \frac{1}{a-1},1 \right)$ through $2\pi $ radians about the $y$-axis. Find the volume of the container, giving your answer as a single fraction in terms of $a$ and $\pi $.[4] Another curved container with a flat circular top is formed in the same way from the curve $x=\frac{\sqrt{y}}{b-{{y}^{2}}}$ and the points $(0,0)$ and $\left( \frac{1}{b-1},1 \right)$. It has a volume that is four times as great as the container in part (i). Find an expression for $b$ in terms of $a$. [3]

Accepted Answer

(a) $\frac{243}{16}$ units$^{2}$ (b)(i) $\frac{\pi }{2a\left( a-1 \right)}$ units$^{3}$ (ii) $b=\frac{1+\sqrt{{{a}^{2}}-a+1}}{2}$

N2017 P2 Q4

N2017 P2 Q4

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