Let's approach the problem step by step.

Step 1: Calculate the volume of water already in the tank

The water is initially filled to a height of 12 cm. The base area of the cuboidal tank is given as 150 square centimeters.

The volume of water initially in the tank is: $\text{Volume of water} = \text{Base area} \times \text{Height of water}$ $\text{Volume of water} = 150 \, \text{cm}^2 \times 12 \, \text{cm} = 1800 \, \text{cm}^3$

Step 2: Calculate the volume of water after immersion

After immersing 6 metal cubes, the water level rises to 17 cm. Thus, the new volume of water in the tank is: $\text{New volume of water} = \text{Base area} \times \text{New height of water}$ $\text{New volume of water} = 150 \, \text{cm}^2 \times 17 \, \text{cm} = 2550 \, \text{cm}^3$

Step 3: Calculate the volume displaced by the cubes

The volume displaced by the 6 metal cubes is the difference between the new volume of water and the initial volume: $\text{Volume displaced} = 2550 \, \text{cm}^3 - 1800 \, \text{cm}^3 = 750 \, \text{cm}^3$

Step 4: Calculate the volume of a single cube

Since 6 solid cubes are immersed in the water, the total volume displaced is equal to the combined volume of the 6 cubes. Let the side length of each cube be $a$ cm. The volume of one cube is: $\text{Volume of one cube} = a^3$ The total volume of 6 cubes is: $6a^3 = 750 \, \text{cm}^3$ Now, solve for $a^3$: $a^3 = \frac{750}{6} = 125 \, \text{cm}^3$ Taking the cube root of both sides: $a = \sqrt[3]{125} = 5 \, \text{cm}$

Final Answer:

The side length of each solid cube is $5$ cm.

Would you like any further details or clarification on the steps? Here are 5 related questions:

1. How do you calculate the volume of a cube with a different side length?
2. What happens to the water level if more cubes are immersed?
3. How can you apply this principle in real-life engineering tasks?
4. What is the formula for the volume of cuboidal tanks with varying base areas?
5. How would the problem change if the tank had a different shape, like a cylinder?

Tip: Always verify that units are consistent when working with volumes, areas, or heights to avoid calculation errors.