A cuboidal shaped water tank of base area 150 squarecentimeters is filled with water up to 12 cm height. After immersing 6 solid metal cubes of side length a cm, in the water of above tank, the water level is goes up to 17cm. Find the side length of a solid cube.

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.

A cuboidal shaped water tank of base area 150 square centimeters is filled with water up to 12 cm height. After immersing 6 solid metal cubes of side length a cm, in the water of above tank, the water level goes up to 17 cm. Find the side length of a solid cube.

Solve a problem involving a cuboidal water tank with base area 150 cm² and initial water height of 12 cm. After immersing 6 solid cubes, the water level rises to 17 cm. Find the side length of each cube using volume displacement principles. Suitable for students in Grades 7-9.

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