Mathematical Analysis: Calculating Water Depth After Ice Melt in a Rectangular Tank

When solving real-world problems involving volume and geometry, it's essential to break down the situation into manageable steps and apply mathematical principles accurately. A common scenario involves determining the new depth of water in a tank after a piece of ice has melted. This article will walk through the process of calculating the new water depth in a rectangular tank when a specific amount of ice melts.

Problem Overview

A rectangular water tank has dimensions of 60 cm by 40 cm by 30 cm. A piece of ice with dimensions 20 cm by 15 cm by 12 cm is dropped into the tank of water. The volume of the ice decreases by 1/10 when it melts. The goal is to determine the new depth of the water in the tank when the ice has completely melted.

Step-by-Step Solution

Step 1: Calculate the Tank's Volume

The volume of the water tank can be calculated using the formula for the volume of a rectangular prism:

Volume of tank length × width × height

Substituting the given dimensions:

Volume of tank 60 cm × 40 cm × 30 cm 72,000 cubic centimeters (cm3)

Step 2: Calculate the Volume of the Ice

The volume of the ice block can be calculated similarly:

Volume of ice 20 cm × 15 cm × 12 cm 3,600 cm3

When the ice melts, its volume decreases by 1/10. To find out the final volume of the melted ice, we perform the following calculation:

Final volume of melted ice 3,600 cm3 - (3,600 cm3 × 1/10) 3,600 cm3 - 360 cm3 3,240 cm3

Step 3: Calculate the Total Volume of Water After Melting

The total volume of water in the tank after the ice melts is the sum of the original volume of water and the volume of the melted ice:

Total volume of water 72,000 cm3 3,240 cm3 75,240 cm3

Step 4: Calculate the New Depth of Water

The new depth of water is calculated by dividing the total volume of water by the base area of the tank (length × width):

New depth of water Total volume of water / (length × width)

Substituting the values:

New depth of water 75,240 cm3 / (60 cm × 40 cm) 75,240 cm3 / 2,400 cm2 31.35 cm

Conclusion

The new depth of the water in the tank is 31.35 cm when the ice has completely melted. This process demonstrates the practical application of volume calculations in real-world scenarios, such as in the operation and maintenance of water tanks.

Additional Insights

Understanding how to calculate the volume and depth of fluid in a tank is crucial for various applications, including but not limited to, water management, industrial processes, and environmental science. The provided steps can be adapted to different dimensions and volumes, making them a valuable tool for problem-solving in various fields.

Keywords

rectangular tank, volume calculation, ice melting, water depth, math problem

References

1. [Reference 1 - Source of Tank Dimensions]

2. [Reference 2 - Methodology for Volume Calculations]

3. [Reference 3 - Application of Volume and Geometry in Real-World Problems]