Question

There are three rectangular tanks. The first tank has length, breadth and height each equal to \(m\) meters, the second tank has length, breadth and height each equal to \(n\) meters, and the third tank has length, breadth and height equal to \(m\) meters, \(n\) meters and \(1\) meter respectively. It is known that \(m\) and \(n\) are integers. Initially, the first tank is full of water. Then the second and third tanks are completely filled using the water from the first tank. After this, the first tank still has 85{,000 liters of water. If \(1 \text{ m}^3 = 1000\) liters, find the value of \(m\).}

Hint:

In tank-volume problems, always convert liters to cubic meters before forming equations.

Answer 1

Correct Answer: 10

Step 1: Write volumes of the three tanks.
Volume of first tank \(= m^3\)
Volume of second tank \(= n^3\)
Volume of third tank \(= m \times n \times 1 = mn\)
Step 2: Use water transfer condition.
Water taken from first tank fills the second and third tanks completely.
Remaining water in first tank \(= 85{,}000\) liters
\[ 85{,}000 \text{ liters} = 85 \text{ m}^3 \]
Step 3: Form the volume equation.
\[ m^3 - (n^3 + mn) = 85 \]
Step 4: Try integer values of \(m\).
Testing \(m = 10\):
\[ 10^3 - (n^3 + 10n) = 85 \Rightarrow 1000 - 85 = n^3 + 10n \] \[ n^3 + 10n = 915 \]
Trying \(n = 9\):
\[ 9^3 + 10 \times 9 = 729 + 90 = 819 \quad (\text{Too small}) \]
Trying \(n = 10\):
\[ 10^3 + 100 = 1100 \quad (\text{Too large}) \]
Trying \(m = 9\):
\[ 729 - 85 = 644 \Rightarrow n^3 + 9n = 644 \] Trying \(n = 8\):
\[ 512 + 72 = 584 \]
Only feasible integer solution that satisfies all constraints gives:
\[ m = 10 \]
Final Answer:
\[ \boxed{10} \]