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To begin, we can work out the volume of
each solid:
Solid 1) A sphere of radius $1 \ \mathrm{cm}$ :
$$\textrm{[Volume]} = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi \
\mathrm{cm^3}$$
Solid 2) A solid cylinder of height $\frac{4}{3} \ \mathrm{cm}$ and
radius $1 \ \mathrm{cm}$:
$$\textrm{[Volume]} = \pi r^2 h = \frac{4}{3}\pi \
\mathrm{cm^3}$$
Solid 3) A solid circular cone of base radius $1 \ \mathrm{cm}$ and
height $4 \ \mathrm{cm}$.
$$\textrm{[Volume]} = \frac{1}{3} \times \textrm{[Volume of a
cylinder]} = \frac{1}{3}\pi r^2 h = \frac{4}{3}\pi \
\mathrm{cm^3}$$
Solid 4) A solid cylinder of height $\frac{4}{9}\ \mathrm{cm}$ with
a hole drilled through it, leaving an annular cross-section with
interal and external radii $2 \ \mathrm{cm}$ and $1 \
\mathrm{cm}$.
$$
\begin{align}
\textrm{[Volume]} & = \textrm{[Volume of the outer cylinder]} -
\textrm{[Volume of the inner cylinder]} \\
& = \pi (r_{outer}^{2} - r_{inner}^{2}) h = \frac{4}{3}\pi
\mathrm{\ cm^3}\\
\end{align}
$$
We can now work out what the axes
represent:
$y$-axis:
The maximum $y$ value reached by all curves is identical at around
4.2, it is in fact equal to $\frac{4}{3}\pi$. This suggests the y
axis is a measure of volume, all solids will eventually displace a
fluid of $\frac{4}{3}\pi \mathrm{\ cm^3}$.
The $y$ axis represents the volume of fluid displaced (or the
volume of the solid immersed) in $\mathrm{cm^3}$.
$x$-axis:
The $x$-axis represents the time elapsed in minutes since lowering
began.
Simon now works out which curve matches
which solid, with some clear reasoning:
Curve 1:
The time taken to fully immerse the object $\approx 1.3 \mathrm{\
minutes}$
The volume displaced varies linearly with time, this must therefore
represent:
Solid 2 (cylinder of height $\frac{4}{3} \mathrm{\ cm}$ and radius
$1 \mathrm{\ cm}$) lowered vertically
or
Solid 4 (A solid cylinder of height $\frac{4}{9} \mathrm{\ cm}$
with a hole drilled through it) lowered vertically.
The time taken to fully immerse the object = $\frac{4}{3}\pi
\mathrm{\ minutes}$
Curve 1 is therefore a cylinder of height $\frac{4}{3} \mathrm{\
cm}$ and radius $1 \mathrm{\ cm}$ lowered vertically
Curve 2 and Curve 3:
The time taken to fully immerse the object = $2 \mathrm{\
minutes}$
This could therefore be:
Solid 2 lowered sideways or solid 1 lowered in any
orientation.
Solid 2 lowered sideways would initially displace a greater fluid
than solid 1, it can therefore be seen that curve 2 corresponds to
solid 2 lowered sideways and curve 3 to solid 1.
Curve 4 and Curve 5:
The time taken to fully immerse the object = $4 \mathrm{\
minutes}$
This could therefore be:
Solid 3 lowered vertically or solid 4 lowered sideways
Consider solid 3 lowered vertically:
Volume immersed as a function of height ($h$)
$$V(h) = \frac{1}{3}\pi r^2 h$$
If we immerse this cone point first the radius varies with the
height of object immersed as:
$$r = \frac{h}{4}$$
$$V(h) = \frac{\pi h^3}{48} \textrm{ or } V(t) = \frac{\pi t^3}{48}
\ (\textrm{as } 1 = \frac{h}{t})$$
Solid 3 immersed point first must therefore be curve 5 which leaves
solid 4 lowered vertically as curve 4.
It is possible to find algebraic forms for
the volume displaced at height h as shown below, although this is
very involved for some of the solids!
Curve 1:
Cylinder of height $\frac{4}{3} \mathrm{\ cm}$ and radius $1
\mathrm{\ cm}$ lowered vertically.
$$V(h) = \pi h$$
Curve 2:
A solid cylinder of height $\frac{4}{3} \mathrm{\ cm}$ and radius
$1 \mathrm{\ cm}$ lowered sideways
The volume immersed equals the length of cylinder multiplied by the
area of a segment
From the geometry it can be seen that $\cos(\frac{\theta}{2}) =
(1-h)$
$\theta = 2\arccos(1-h)$
$\textrm{[Area of segment]} = 0.5(r^2\theta -r^2 \sin(\theta)) =
0.5(2\arccos(1-h) - \sin(2\arccos(1-h))$
$\sin\theta = 2 \sin\frac{\theta}{2} \cos\frac{\theta}{2}$
$\sin\frac{\theta}{2} = \sqrt{1 - \cos^2\frac{\theta}{2}}$
$\sin(2 \arccos(1-h)) = 2\sqrt{1 - (1-h)^2} (1-h) $
$\textrm{[Area of segment]} = 0.5(r^2\theta -r^2 \sin(\theta)) =
0.5(2\arccos(1-h) - \sin(2\arccos(1-h))$
$\textrm{[Area of segment]} = \frac{1}{2}( 2\arccos(1-h) - 2\sqrt{1
- (1-h)^2}) (1-h)$
The volume immersed equals the length of the cylinder multiplied by
the area of the segment
$\textrm{[Volume at h]} = \frac{2}{3}( 2\arccos(1-h) - 2\sqrt{1 -
(1-h)^2}) (1-h)$
Curve 3:
A sphere of radius $1 \mathrm{\ cm}$
The volume immersed as a function of h is equal to the volume
generated when we rotate the equation of a circle about the
$x$-axis by 360 degrees and evaluate this integral between the
limits $r$ and $(r-h)$.
Equation of circle: $y^2 + x^2 = r^2 = 1$
$$\textrm{[Volume]} = \int_{1-h}^{1} \pi f(x)^2 \ \mathrm{d}x =
\int_{1-h}^{1} \pi (1-x^2) \ \mathrm{d}x =\pi \left[ x -
\frac{x^3}{3} \right]^{1} _{1-h} = \pi h^2(1-\frac{h}{3})$$
Curve 4:
A solid cylinder of height $\frac{4}{9} \mathrm{\ cm}$ with a hole
drilled through it, leaving an annular cross-section with interal
and external radii $2 \mathrm{\ cm}$ and $1 \mathrm{\ cm}$ lowered
sideways.
When $h$ is less than $1 \mathrm{\ cm}$ the volume immersed takes
the same form as curve 2 but simply changing the radius from $1
\mathrm{\ cm}$ to $2 \mathrm{\ cm}$ and changing the length of the
solid from $\frac{4}{3} \mathrm{\ cm}$ to $\frac{4}{9} \mathrm{\
cm}$.
From the geometry we see that:
$\cos(\frac{\theta}{2}) = \frac{2 - h}{2}$
$\theta = 2 \arccos(1-0.5h)$
$A(h) = 0.5(r^2\theta -r^2 \sin\theta) = 0.5(8 \arccos(1- 0.5h) - 4
\sin (2 \arccos(1-0.5h))$
$\sin\theta = 2\sin\frac{\theta}{2} \cos\frac{\theta}{2}$
$\sin\frac{\theta}{2} = \sqrt{1 - \cos^2\frac{\theta}{2}}$
$\sin\theta = 2\sqrt{1 - (1-0.5h)^2} (1-0.5h) $
$\therefore \ A(h) = 4 \arccos(1- 0.5h) - 4(1-0.5h)\sqrt{1 -
(1-0.5h)^2}$
$\textrm{[Volume]} = \frac{4}{9} \textrm{[area]} =
\frac{16}{9}(\arccos(1- 0.5h) - (1-0.5h)\sqrt{1 -
(1-0.5h)^2}$
(when $h$ is less than 1)
When h is greater than $1\mathrm{\ cm}$ the volume immersed can be
found by subtracting the area of the inner segment from the outer
segment and then multiplying by the length of the cylinder.
Volume of the outer segment is as above:
$\textrm{[Volume]} = \frac{4}{9} \textrm{[area]} =
\frac{16}{9}(\arccos(1- 0.5h) - (1-0.5h)\sqrt{1 - (1-0.5h)^2}
)$
Volume of inner segment:
From the geometry it can be seen that:
$\theta = 2\arccos(2-h)$
$\textrm{[Volume]} = \frac{4}{9} (0.5(r^2\theta -r^2 \sin\theta)) =
\frac{2}{9} ( 2\arccos(2-h) - \sqrt{1 -(2-h)^2})$
Therefore:
$
\begin{align}
\textrm{[Volume immersed]} & = \frac{4}{9}(4\arccos(1- 0.5h) -
4(1-0.5h)\sqrt{1 - (1-0.5h)^2)} \\
& -\arccos(2-h) + 0.5 \sqrt{1 -(2-h)^2} )\\
\end{align}
$
(when $h$ is greater than 1)
Therefore
(when $h$ is less than 1)
$V(h) = \frac{4}{9} \textrm{[area]} = \frac{16}{9}(\arccos(1- 0.5h)
- (1-0.5h)\sqrt{1 - (1-0.5h)^2} )$
(when $h$ is greater than 1)
$V(h) = \frac{4}{9}(4\arccos(1- 0.5h) - 4(1-0.5h)\sqrt{1 -
(1-0.5h)^2)} -\arccos(2-h) + 0.5 \sqrt{1 -(2-h)^2} )$
Curve 5:
A solid circular cone of base radius $1 \mathrm{\ cm}$ and height
$4 \mathrm{\ cm}$ lowered point first.
$V(h) = \frac{1}{3}\pi r^2 h$
$r(h) = \frac{h}{4}$
$V(h) = \frac{\pi h^3}{48} $