Alan

The following images might help:

So sin θ ≈ θ only when θ is small

Does the following help?

Use the calculator here to find out:

$$\underset{\,\,\,\,{\textcolor[rgb]{0.66,0.66,0.66}{\rightarrow {\mathtt{x}}}}}{{solve}}{\left(\begin{array}{l}{\mathtt{a}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{c}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{d}}={\mathtt{0}}\end{array}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = \left({\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{d}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}\left({\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}\right){\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{2}}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,-\,}}{\frac{\left({\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,-\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}\right)}{\left({\mathtt{54}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}\right)}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\frac{\left({\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}\right)}{\left({\mathtt{9}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{d}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}\left({\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}\right){\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{2}}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,-\,}}{\frac{\left({\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,-\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}\right)}{\left({\mathtt{54}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}\right)}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,-\,}}{\frac{{\mathtt{b}}}{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{a}}\right)}}\\
{\mathtt{x}} = \left({\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{d}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}\left({\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}\right){\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{2}}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,-\,}}{\frac{\left({\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,-\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}\right)}{\left({\mathtt{54}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}\right)}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\frac{\left({\mathtt{\,-\,}}{\frac{{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}\right)}{\left({\mathtt{9}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{d}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}\left({\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}\right){\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{2}}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,-\,}}{\frac{\left({\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,-\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}\right)}{\left({\mathtt{54}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}\right)}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,-\,}}{\frac{{\mathtt{b}}}{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{a}}\right)}}\\
{\mathtt{x}} = {\left({\frac{{\sqrt{{\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{d}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}\left({\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}\right){\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{2}}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,-\,}}{\frac{\left({\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,-\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}\right)}{\left({\mathtt{54}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}\right)}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,-\,}}{\frac{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}\right)}{\left({\mathtt{9}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\left({\frac{{\sqrt{{\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{d}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}\left({\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}\right){\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,\small\textbf+\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{3}}}{\mathtt{\,-\,}}{{\mathtt{b}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{2}}}}}}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{\left({\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,-\,}}{\frac{\left({\mathtt{27}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{d}}{\mathtt{\,-\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{a}}{\mathtt{\,\times\,}}{\mathtt{b}}{\mathtt{\,\times\,}}{\mathtt{c}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{b}}}^{{\mathtt{3}}}\right)}{\left({\mathtt{54}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{3}}}\right)}}\right)}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}}{\mathtt{\,-\,}}{\frac{{\mathtt{b}}}{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{a}}\right)}}\\
\end{array} \right\}$$

The Klein bottle is a 4-D object!  The physical ones you can see are just projections of the 4-D version in 3-D.

Think of this as a function t(n) = n - 1

t(1) = 0

t(2) = 1

t(3) = 2

...

t(10) = 9

t(11) = 10

...

t(99) = 98

t(100) = 99

Clearly t(100) = 99 = 98 + 1 = t(99) + 1

Also t(11)² - 1 = 10² -1 = 99 = t(100) = t(99) + 1

I'm not sure I'd classify this as one of life's great mysteries!!

All that needs to be done to find the maximum and minimum depths is to note that sin has a maximum of 1 and a minimum of -1, so the maximum depth will be 4.5 + 1.5 = 6, and the minimum will be 4.5 - 1.5 = 3.

If the times at which they occur are needed then we can just set pi*t/12 = pi/2 or t=6 (and every 24 hours (?) thereafter) for the maximum and pi*t/12 = 3pi/2 or t = 18 (and every 24 hours thereafter) for the minimum.

The calculus approach, though perfectly valid, is overkill here!

I think the Math Formula tab was used to enter this question.  See for example:

$$\mathrm{ERROR}{\mathtt{\,-\,}}{\mathtt{22}}{\mathtt{\,-\,}}{\mathtt{22}}{\mathtt{\,-\,}}{\mathtt{20}}$$

I used the Math Formula tab and entered: find the mean -22 -22 -20 to get this. The Math Formula tab doesn't like spaces between words; it thinks words are variable names.

First do 45 + 9 = 54, then add 1/2 + 2/3 

$$\frac{1}{2}+\frac{2}{3}=\frac{3}{6}+\frac{4}{6}=\frac{7}{6}=1\frac{1}{6}$$

$$45\frac{1}{2}+9\frac{2}{3}= 55\frac{1}{6}$$

A Klein bottle has no interior and no exterior so it has zero volume!