Lesson A partially filled cylindrical tank changes position from horizontal to vertical

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<H2>A partially filled cylindrical tank changes position from horizontal to vertical</H2>

<H3>Problem 1</H3>A closed cylindrical tank is &nbsp;8 &nbsp;feet long and &nbsp;3 &nbsp;feet in diameter. 
When lying in a horizontal position, &nbsp;the water is &nbsp;2 &nbsp;feet deep. 
If the tank is the vertical position, &nbsp;find the depth of water in the tank.

<B>Solution</B>

<pre>
The formula for the volume of water in horizontal cylindrical tank is

V = {{{L*(r^2*arccos((r-h)/r) - (r-h)*sqrt(2rh-h^2)))}}},    (1)

where 'r' is the radius of the cylindrical tank and 'h' is the depth of water; L is the length of the cylinder.

This formula represents the product of the length of the container by the area of the cross-section
of the tank, occupied by water.

Notice that in this problem the depth 'h' of 2 feet is greater than the radius of the tank, which is 1.5 ft.

So, the horizontal axis of the container is BELOW the water level.

Nevertheless, the formula works in this case too, without change.

Indeed, when h > r, the first term represents the area of the major sector of the circle, 
while the second term represents the area of the triangle, which complement the sector to the major segment.

So, we are ready to calculate. Insert the numbers instead of symbols

V = {{{8*(1.5^2*arccos((1.5-2)/1.5) - (1.5-2)*sqrt(2*1.5*2-2^2))}}}.    (2)

We have  {{{arccos((1.5-2)/1.5)}}} = {{{arccos(-1/3)}}} = 1.910633 radians,

{{{(1.5-2)*sqrt(2*1.5*2-2^2)}}} = {{{-0.5*sqrt(2)}}} = -0.707107.

so we can continue formula (2) this way

V = {{{8*(1.5^2*1.910633 - (-0.707107))}}} = 40.04825 ft^3.

Now, to get the height of the water in vertical cylinder, we should divide this volume
by the area of the base  {{{pi*r^2}}} = {{{3.14159*1.5^2}}} = 7.0685775.

Thus we find

the height of the water in vertical container = {{{40.04825/7.0685775}}} = 5.665673185

or about 5.666 ft.

<U>ANSWER</U>.  The height of the water in vertical container is about 5.666 ft.
</pre>

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