<p>The curve $C$ is defined by the parametric equations</p>
<p style="text-align: center">$x=6t-1$, $y=8{{t}^{2}}+2$ where $t\ge \frac{1}{6}$.</p>
<p>The line $N$, with equation $3y=8x-10$, is the tangent to $C$ at the point $\left( 5,10 \right)$. The region enclosed by $C$, $N$ and the $y$-axis is denoted by $R$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the area of $R$.</div></li><li><div class="list-item-paragraph">Find the cartesian equation of $C$. Hence, find the volume generated when $R$ is rotated about the $y$-axis through ${{360}^{\circ }}$.</div></li></ol>

The curve $C$ is defined by the parametric equations

$x=6t-1$, $y=8{{t}^{2}}+2$ where $t\ge \frac{1}{6}$.

The line $N$, with equation $3y=8x-10$, is the tangent to $C$ at the point $\left( 5,10 \right)$. The region enclosed by $C$, $N$ and the $y$-axis is denoted by $R$.

Find the cartesian equation of $C$. Hence, find the volume generated when $R$ is rotated about the $y$-axis through ${{360}^{\circ }}$.

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