(ii) $x^{\frac{2}{3}}+y^{\frac{2}{3}}=\left(\frac{a}{2}\right)^{\frac{2}{3}}$

(iii)$0.37 a^2$

(i)

Let the tangent to $C$ at the point $P\left(a \sin ^5 t, a \cos ^5 t\right)$ cut the $x$ and $y$-axes at $A$ and $B$ respectively. $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} / \frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{-5 a \sin t \cos ^4 t}{5 a \cos t \sin ^4 t}=-\cot ^3 t$
Equation of tangent at $P$ is
$y-a \cos ^5 t=-\cot ^3 t\left(x-a \sin ^5 t\right)$
At $A, y=0$. Substituting into equation of tangent gives
$$\begin{aligned}
x &=a \sin ^5 t+\frac{a \cos ^5 t}{\cot ^3 t} \\
&=a \sin ^5 t+a \sin ^3 t \cos ^2 t \\
&=a \sin ^3 t\left(\sin ^2 t+\cos ^2 t\right) \\
&=a \sin ^3 t
\end{aligned}$$

At $B, x=0$. Substituting into equation of tangent gives
$$\begin{aligned}
y &=a \cos ^5 t+a \sin ^5 t \cot ^3 t \\
&=a \cos ^5 t+a \sin ^2 t \cos ^3 t \\
&=a \cos ^3 t\left(\cos ^2 t+\sin ^2 t\right) \\
&=a \cos ^3 t \\
O A^{\frac{2}{3}}+O B^{\frac{2}{3}} &=\left(a \sin ^3 t\right)^{\frac{2}{3}}+\left(a \cos ^3 t\right)^{\frac{2}{3}} \\
&=a^{\frac{2}{3}}\left(\sin ^2 t+\cos ^2 t\right)=a^{\frac{2}{3}}
\end{aligned}$$

(ii) At $B, x=0$. Substituting into equation of tangent gives
$$\begin{aligned}
y &=a \cos ^5 t+a \sin ^5 t \cot ^3 t \\
&=a \cos ^5 t+a \sin ^2 t \cos ^3 t \\
&=a \cos ^3 t\left(\cos ^2 t+\sin ^2 t\right) \\
&=a \cos ^3 t \\
O A^{\frac{2}{3}}+O B^{\frac{2}{3}} &=\left(a \sin ^3 t\right)^{\frac{2}{3}}+\left(a \cos ^3 t\right)^{\frac{2}{3}} \\
&=a^{\frac{2}{3}}\left(\sin ^2 t+\cos ^2 t\right)=a^{\frac{2}{3}}
\end{aligned}$$

(iii)The curve $C$ is symmetrical about the $x$ and $y$ axes. Area enclosed by $C$
$$
\begin{array}{l}
=4 \int_0^a y \mathrm{~d} x \\
=4 \int_0^{\frac{\pi}{2}} a \cos ^5 t\left(5 a \sin ^4 t \cos t\right) \mathrm{d} t \\
=20 a^2 \int_0^{\frac{\pi}{2}} \sin ^4 t \cos ^6 t \mathrm{~d} t \\
\approx 0.37 a^2
\end{array}
$$