The data shows volume and height of 4 right circular tanks P, Q, R and S. The given data also depicts time taken to fill or empty the tank by inlet pipe A, B and outlet pipe C.

1. Ratio between time taken by pipe B and that by pipe C is different because pipes were used by the operator in different ways for different tanks.
2. Radius of tank S is 10.5 m.

Tank	Volume	Height	Time taken by pipe A to fill tank (in hours)	Ratio of time taken by pipe B to fill and pipe C to empty the tank
P	38808	28	10	2: 3
Q	--------	10	12	3: 2
R	5390	35	18	1: 3
S	19250	--------	---------	2: 5

In tank R, pipe A was opened for 9 hours and then closed. Pipe B and C were the opened together for 3 hours and then both pipes were closed. After a total of 12 hours, approximately what volume of tank remained empty (in cubic meters), if both pipes A and C were opened together, tank will always be empty).

First, let us analyze the data given for Tank R:

• Total Volume of Tank R = 5390 m³
• Time taken by Pipe A to fill Tank R = 18 hours
• Ratio of time taken by Pipe B to fill and Pipe C to empty Tank R = 1 : 3

Step 1: Calculate the volume filled by Pipe A in 9 hours
Since Pipe A takes 18 hours to fill the entire tank, the fraction of the tank filled by Pipe A in 9 hours is:
$\backslash text\{Fraction\; filled\}\; =\; \backslash frac\{9\backslash text\{\; hours\}\}\{18\backslash text\{\; hours\}\}\; =\; \backslash frac\{1\}\{2\}$

The volume filled by Pipe A during this time is:
$\backslash text\{Volume\; filled\; by\; A\}\; =\; 5390\backslash text\{\; m\}^3\; \backslash times\; \backslash frac\{1\}\{2\}\; =\; 2695\backslash text\{\; m\}^3$

Step 2: Determine the capacities and rates of Pipe B and Pipe C
Let the time taken by Pipe B to fill the tank be x hours.
Since the ratio of time taken by B to C is 1 : 3, the time taken by Pipe C to empty the tank is 3x hours.

The problem states: "if both pipes A and C were opened together, tank will always be empty".
This condition implies that the emptying rate of Pipe C must be greater than or equal to the filling rate of Pipe A:
$\backslash text\{Rate\; of\; C\}\; \backslash ge\; \backslash text\{Rate\; of\; A\}$
$\backslash frac\{1\}\{3x\}\; \backslash ge\; \backslash frac\{1\}\{18\}\; \backslash Rightarrow\; 3x\; \backslash le\; 18\; \backslash Rightarrow\; x\; \backslash le\; 6\backslash text\{\; hours\}$

Taking the limiting case for maximum efficiency under the given constraints:
$x\; =\; 6\backslash text\{\; hours\}$

Therefore, Pipe B takes 6 hours to fill the tank, and Pipe C takes 18 hours to empty the tank.

Step 3: Calculate the volume filled by Pipes B and C in 3 hours
When Pipes B and C are opened together, the net filling rate per hour is:
$\backslash text\{Net\; Rate\}\; =\; \backslash frac\{1\}\{\backslash text\{Time\; taken\; by\; B\}\}\; -\; \backslash frac\{1\}\{\backslash text\{Time\; taken\; by\; C\}\}\; =\; \backslash frac\{1\}\{6\}\; -\; \backslash frac\{1\}\{18\}$
$\backslash text\{Net\; Rate\}\; =\; \backslash frac\{3\; -\; 1\}\{18\}\; =\; \backslash frac\{2\}\{18\}\; =\; \backslash frac\{1\}\{9\}\backslash text\{\; of\; the\; tank\; per\; hour\}$

In 3 hours, the fraction of the tank filled by both pipes working together is:
$\backslash text\{Fraction\; filled\}\; =\; \backslash frac\{1\}\{9\}\; \backslash times\; 3\; =\; \backslash frac\{1\}\{3\}\backslash text\{\; of\; the\; tank\}$

The volume filled by Pipes B and C in 3 hours is:
$\backslash text\{Volume\; filled\; by\; B\; and\; C\}\; =\; 5390\backslash text\{\; m\}^3\; \backslash times\; \backslash frac\{1\}\{3\}\; \backslash approx\; 1796.67\backslash text\{\; m\}^3$

Step 4: Calculate the remaining empty volume of the tank
The total volume filled after 12 hours is:
$\backslash text\{Total\; filled\; volume\}\; =\; 2695\backslash text\{\; m\}^3\; +\; 1796.67\backslash text\{\; m\}^3\; =\; 4491.67\backslash text\{\; m\}^3$

The volume of the tank that remains empty is:
$\backslash text\{Empty\; volume\}\; =\; 5390\backslash text\{\; m\}^3\; -\; 4491.67\backslash text\{\; m\}^3\; =\; 898.33\backslash text\{\; m\}^3\; \backslash approx\; 900\backslash text\{\; m\}^3$

Therefore, the volume of the tank that remained empty is approximately 900 m³.