Cube-shaped Tank, Volume Calculation, Side Length, Water Capacity, Filling The Tank

This comprehensive guide provides a detailed solution to a classic problem involving a cube-shaped water tank. We will break down each step, ensuring clarity and understanding. This step-by-step solution is designed to guide you through the problem-solving process, making it easy to grasp the underlying concepts and apply them to similar scenarios.

Problem Statement

A cube-shaped tank contains 16200 liters of water. Knowing that the water in the tank represents 3/5 of the tank's capacity, calculate:

a) The volume of the cube. b) The length of the side of the cube. c) How many liters of water need to be added to the tank to fill it completely.

Solution

a) Calculating the Volume of the Cube

The first step in solving this problem is to determine the total volume of the cube-shaped tank. We know that the current water level, which is 16200 liters, represents 3/5 of the total capacity. To find the full volume, we need to figure out what 1/5 of the capacity is and then multiply that by 5. This will give us the total volume of the tank.

Determining the Fraction Representation: Understanding fractions is crucial here. The fraction 3/5 means that the tank is divided into five equal parts, and the water currently fills three of those parts. The remaining two parts are empty, and our goal is to find the capacity of all five parts, which represents the tank's full volume. To find the total capacity, we start by determining the value of one of these parts.

Finding 1/5 of the Capacity: Since 16200 liters corresponds to 3/5 of the tank's capacity, we can find 1/5 of the capacity by dividing 16200 liters by 3. This will give us the amount of water that fills one-fifth of the tank. So, we perform the calculation: 16200 liters / 3 = 5400 liters. This means that each 1/5 portion of the tank can hold 5400 liters of water.

Calculating the Total Volume: Now that we know 1/5 of the capacity is 5400 liters, we can find the total volume (which is 5/5) by multiplying 5400 liters by 5. This calculation will give us the total capacity of the tank. So, we calculate: 5400 liters * 5 = 27000 liters. Therefore, the total volume of the cube-shaped tank is 27000 liters.

Volume in Cubic Decimeters: It's important to note that 1 liter is equivalent to 1 cubic decimeter (dm³). This conversion is crucial for the next part of the problem, where we need to find the side length of the cube. Since we now know the volume in liters, we also know it in cubic decimeters. Thus, the volume of the cube is 27000 dm³.

In summary, to find the volume of the cube, we used the information that 3/5 of the tank's capacity is 16200 liters. By dividing 16200 by 3, we found 1/5 of the capacity, which is 5400 liters. Multiplying this by 5 gave us the total volume of 27000 liters, which is equivalent to 27000 cubic decimeters.

b) Determining the Length of the Side of the Cube

Having calculated the volume of the cube as 27000 cubic decimeters, our next step is to find the length of one side of the cube. The formula for the volume of a cube is V = s³, where V is the volume and s is the length of a side. To find s, we need to calculate the cube root of the volume.

Understanding the Cube Root: The cube root of a number is a value that, when multiplied by itself three times, equals the original number. In mathematical terms, if we have a number x, the cube root of x is a value y such that y * y * y = x. For example, the cube root of 8 is 2 because 2 * 2 * 2 = 8. Finding the cube root is the inverse operation of cubing a number, and it is essential for solving problems involving volumes of cubes.

Calculating the Cube Root of 27000: We need to find a number that, when multiplied by itself three times, equals 27000. To do this, we can break down 27000 into its prime factors or recognize it as a multiple of a known cube. The number 27000 can be expressed as 27 * 1000. We know that 27 is 3 cubed (3 * 3 * 3 = 27) and 1000 is 10 cubed (10 * 10 * 10 = 1000). Therefore, 27000 is (3 * 10) cubed, which means 27000 = 30³.

So, the cube root of 27000 is 30. This can be written mathematically as: √3 = 30. This result is crucial because it tells us the length of one side of the cube.

Interpreting the Result: The cube root of 27000 being 30 means that the side length (s) of the cube is 30 decimeters. Since we know that 1 decimeter is equal to 10 centimeters, we can also express the side length in centimeters. To convert decimeters to centimeters, we multiply by 10. Thus, 30 decimeters is equal to 30 * 10 = 300 centimeters.

Summary of the Side Length Calculation: In summary, we started with the volume of the cube, which is 27000 cubic decimeters. To find the side length, we calculated the cube root of the volume. The cube root of 27000 is 30, meaning the side length of the cube is 30 decimeters. We also converted this length to centimeters, finding that the side length is 300 centimeters. This result gives us a clear understanding of the physical dimensions of the cube-shaped tank.

c) Determining the Additional Water Needed to Fill the Tank

In this part of the problem, we need to calculate how much more water is required to completely fill the tank. We know that the tank currently holds 16200 liters of water, and the total capacity of the tank is 27000 liters. To find out how much more water is needed, we will subtract the current water level from the total capacity.

Understanding the Problem: The problem essentially asks us to find the difference between the full capacity of the tank and the amount of water already in it. This difference represents the empty space in the tank, which needs to be filled. To calculate this, we will perform a simple subtraction operation.

Performing the Subtraction: We subtract the current water level (16200 liters) from the total capacity (27000 liters). This can be written as: 27000 liters - 16200 liters. Performing this subtraction gives us the amount of water needed to fill the tank completely.

The calculation is as follows:

27000

• 16200

10800

Therefore, 27000 liters - 16200 liters = 10800 liters.

Interpreting the Result: The result of the subtraction, 10800 liters, tells us exactly how much additional water is needed to fill the tank to its full capacity. This means that if we add 10800 liters of water to the tank, it will be completely full.

Contextualizing the Answer: Knowing that we need to add 10800 liters of water gives us a practical understanding of the situation. This quantity represents the remaining 2/5 of the tank's capacity, as the tank is currently 3/5 full. Adding this amount will bring the water level up to the tank's maximum capacity.

Summary of the Calculation: In summary, we calculated the additional water needed to fill the tank by subtracting the current water level (16200 liters) from the total capacity (27000 liters). The result of this subtraction, 10800 liters, is the amount of water that needs to be added to completely fill the tank. This concludes the solution for part (c) of the problem, providing a clear and concise answer to the question.

Final Answers

a) The volume of the cube is 27000 liters (or 27000 dm³). b) The length of the side of the cube is 30 decimeters (or 300 centimeters). c) 10800 liters of water need to be added to the tank to fill it completely.

This step-by-step solution provides a clear understanding of how to solve this type of problem. By breaking down each part and explaining the logic behind the calculations, we have made the process easy to follow and understand. This approach can be applied to similar problems involving volumes and capacities, making it a valuable learning tool.