For real numbers $t$ satisfying $-1 \leq t \leq 1$, let $$x(t) = (1+t)\sqrt{1+t}, \quad y(t) = 3(1+t)\sqrt{1-t}$$ Consider the point $\mathrm{P}(x(t),\ y(t))$ in the coordinate plane.
(1) Show that the function $\dfrac{y(t)}{x(t)}$ of $t$ on $-1 < t \leq 1$ is strictly decreasing.
(2) Let $f(t)$ be the distance from the origin to $\mathrm{P}$. Investigate the monotonicity of the function $f(t)$ of $t$ on $-1 \leq t \leq 1$, and find its maximum value.
(3) Let $C$ be the locus of $\mathrm{P}$ as $t$ ranges over $-1 \leq t \leq 1$, and let $D$ be the region enclosed by $C$ and the $x$-axis. When $D$ is rotated $90°$ clockwise about the origin, find the area of the region swept out by $D$. %% Page 4
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\noindent For real numbers $t$ satisfying $-1 \leq t \leq 1$, let
$$x(t) = (1+t)\sqrt{1+t}, \quad y(t) = 3(1+t)\sqrt{1-t}$$
Consider the point $\mathrm{P}(x(t),\ y(t))$ in the coordinate plane.
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\noindent (1) Show that the function $\dfrac{y(t)}{x(t)}$ of $t$ on $-1 < t \leq 1$ is strictly decreasing.
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\noindent (2) Let $f(t)$ be the distance from the origin to $\mathrm{P}$. Investigate the monotonicity of the function $f(t)$ of $t$ on $-1 \leq t \leq 1$, and find its maximum value.
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\noindent (3) Let $C$ be the locus of $\mathrm{P}$ as $t$ ranges over $-1 \leq t \leq 1$, and let $D$ be the region enclosed by $C$ and the $x$-axis. When $D$ is rotated $90°$ clockwise about the origin, find the area of the region swept out by $D$.
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