Question
Consider the function \(f(x)=\frac{e^x-e^{-x}}{2}, x \in \mathbb{R}\).
(a) Show that \(f\) is an odd function.
Now, consider the function \(g\) given by \(g(x)=\frac{x^4+2}{2 x}, x \in \mathbb{R}, x \neq 0\).
(b) By considering the graph of \(y=f(x)-g(x)\), solve \(f(x)>g(x)\) for \(x \in \mathbb{R}\).
Question
The following diagram shows the curve \(\frac{x^2}{400}+\frac{(y+5)^2}{225}=1\), where \(0 \leq y \leq h\).
The curve from point \(\mathrm{C}\) to point \(\mathrm{P}\) is rotated \(360^{\circ}\) about the \(y\)-axis to form a lamp shade. The rectangle \(\mathrm{ABCD}\), of height \((10-h) \mathrm{cm}\), is rotated \(360^{\circ}\) about the \(y\)-axis to form a solid ceiling fixture.
The lamp shade is assumed to have a negligible thickness.
Given that the interior volume of the lamp shade is to be \(600 \mathrm{~cm}^3\), determine the height of the ceiling fixture, length \(A D\) in the diagram.