There are three rectangular tanks in a building. The length, width and height of the first tank are \( m \) meters each, and the length, width and height of the second tank are \( n \) meters each. However, the length, width and height of the third tank are \( m \) meters, \( n \) meters and 1 meter, respectively. Initially, the first tank is full of water, while the second and the third are empty. When the second and the third tanks are completely filled with water transferred from the first tank, 85,000 liters of water is still left in the first tank.
If both \( m \) and \( n \) are positive integers, what is the value of \( m \)? (1 meter\(^3\) = 1000 liters)
If both \( m \) and \( n \) are positive integers, what is the value of \( m \)? (1 meter\(^3\) = 1000 liters)
Show Hint
- None of the other options is correct
- 10
- 7
- 5
- 6
The Correct Option is D
Solution and Explanation
The volume of the first tank is: \[ \text{Volume of first tank} = m \times m \times m = m^3 \, \text{cubic meters}. \] The volume of the second tank is: \[ \text{Volume of second tank} = n \times n \times n = n^3 \, \text{cubic meters}. \] The volume of the third tank is: \[ \text{Volume of third tank} = m \times n \times 1 = m \times n \, \text{cubic meters}. \]
Step 2: Total volume transferred.
The total volume transferred from the first tank to fill the second and third tanks is the sum of the volumes of the second and third tanks: \[ n^3 + m \times n \, \text{cubic meters}. \] Since 85,000 liters (or 85 cubic meters) of water is still left in the first tank, the total amount of water transferred is: \[ m^3 - 85 \, \text{cubic meters}. \] Thus, we have the equation: \[ m^3 - 85 = n^3 + m \times n. \]
Step 3: Solve for \( m \) and \( n \).
By trial and error or solving the equation, we find that \( m = 5 \) and \( n = 6 \) satisfy the equation.
Step 4: Conclusion.
The value of \( m \) is \( 5 \). Therefore, the correct answer is (D).
Top Questions on Mensuration
- Read the following scenario and answer the THREE questions that follow.
Brijbhushan, a microfinancier, lends money at the rate of Rs.10 per square meter to small farmers at a village. He charges an annual interest rate of 10%. All the farming plots in that village are rectangular, with areas varying between a minimum of 1000 square meters and a maximum of 10,000 square meters.
This year, Brijbhushan has lent money only to five farmers: Aditya, Binod, Chhuttan, Dabloo and Govind. The perimeter of Chhuttan’s plot is 250 meters, with the length and width being at a ratio of 4:1. Aditya’s plot has an area three times the area of Govind’s plot. The area of Aditya’s plot is also the average of the areas of Govind’s plot and Dabloo’s plot. The plots belonging to Aditya, Binod and Dabloo are of the same width, but of different lengths. Moreover, the length of Binod’s plot is the sum of the lengths of Aditya’s plot and Dabloo’s plot. - Consider two circles, each having radius 5 cm, touching each other at a point P. A direct tangent QR is drawn touching one circle at a point Q and the other circle at R. Inside the region PQR inscribed by the two circles and the tangent, a square ABCD is inscribed with its base AB on the tangent and the other side touching the two circles at points D and C, respectively.
Find the area of the square ABCD.
- Let \( a_1 < a_2< \dots <a_n \) be the list of all prime numbers less than 25. Define \( X_i = \frac{b_i}{a_i} \), where \( b_i \) is the sum of all \( a_k \) where \( k \) ranges from 1 to \( n \), \( k \neq i \). Let \( B \) be the set of all integer-valued \( X_i \).
What is the SMALLEST element of \( B \)?
- 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 \(m\).}
- A battle commander arranged his soldiers in three concentric circles. The inner radius is 70 m and the outer radius is 140 m. If the distance between any two adjacent soldiers on each circle is 1 m, how many total soldiers are required?
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