https://file.aiquickdraw.com/mathbot/file/17270268989015vkhvanc.jpg

Let's go through the questions from the image and solve them one by one.

### SECTION-A

1. **Find the cube root of the following number.**
   \[
   140 \times 2450
   \]
   First, calculate \( 140 \times 2450 = 343000 \), and then find the cube root:
   \[
   \sqrt[3]{343000}
   \]

2. **Evaluate**
   \[
   \sqrt[3]{1000} + \sqrt[3]{0.008} + \sqrt[3]{0.125}
   \]
   \[
   \sqrt[3]{1000} = 10, \quad \sqrt[3]{0.008} = 0.2, \quad \sqrt[3]{0.125} = 0.5
   \]
   Therefore, \( 10 + 0.2 + 0.5 = 10.7 \).

---

### SECTION-B

3. **Find the side of a cube whose surface area is 150 sq m.**
   The surface area \( A \) of a cube is given by:
   \[
   A = 6a^2
   \]
   where \( a \) is the side of the cube. Solving for \( a \):
   \[
   6a^2 = 150 \implies a^2 = 25 \implies a = 5 \text{ meters}
   \]

4. **What is the smallest number by which 243000 must be divided so that the quotient is a perfect cube?**
   Factorize 243000 and find the smallest number to divide it such that the quotient becomes a perfect cube.

---

### SECTION-C

5. **How many square meters of cardboard will be needed to make a cube of volume 216 m³?**
   The volume \( V \) of a cube is:
   \[
   V = a^3
   \]
   Solving for \( a \):
   \[
   a = \sqrt[3]{216} = 6 \text{ meters}
   \]
   The surface area of the cube is:
   \[
   A = 6a^2 = 6 \times 6^2 = 216 \text{ square meters}
   \]

6. **Three numbers are in the ratio 2:3:4. The sum of their cubes is 33957. Find the numbers.**
   Let the numbers be \( 2x \), \( 3x \), and \( 4x \). Then:
   \[
   (2x)^3 + (3x)^3 + (4x)^3 = 33957
   \]
   Solve for \( x \) to find the numbers.

---

### SECTION-D

7. **A cuboid of dimensions 126m, 140m, 525m has been melted to form a cube. Find the side of the cube.**
   The volume of the cuboid is:
   \[
   V = 126 \times 140 \times 525
   \]
   The side of the cube is the cube root of the volume:
   \[
   \text{Side of the cube} = \sqrt[3]{126 \times 140 \times 525}
   \]

8. **Prove that if a number is doubled, then its cube is eight times the cube of the original number.**
   Let the original number be \( x \). The cube of the original number is:
   \[
   x^3
   \]
   If the number is doubled, the cube of the new number is:
   \[
   (2x)^3 = 8x^3
   \]
   Hence, the cube of the doubled number is 8 times the cube of the original number.

---

Let me know if you want any detailed steps or explanations for any particular question. 

Here are 5 related questions:

1. What is the cube root of 729000?
2. Find the cube of 12.
3. If the surface area of a cube is 96 square meters, what is its side length?
4. If a cuboid has dimensions 100m, 150m, and 200m, what is its volume?
5. What is the cube of \( 1.5 \)?

**Tip:** Always verify your answers by recalculating key steps to avoid small errors.

Find the cube root of the following number 140x2450. Evaluate ∛1000 + ∛0.008 + ∛0.125. Find the side of a cube whose surface area is 150 sq m. What is the smallest number by which 243000 must be divided so that the quotient is a perfect cube? How many square meters of cardboard will be needed to make a cube of volume 216 m³? Three numbers are to one another as 2:3:4. The sum of their cubes is 33957. Find the numbers. A cuboid of dimensions 126m, 140m, 525m has been melted to form a cube. Find the side of the cube. Prove that if a number is doubled then its cube is eight times the cube of the original number.

Solve a range of cube and cuboid-related problems, including finding cube roots, calculating surface area, determining perfect cubes, and working with ratios. These problems involve applying key formulas for surface area and volume of cubes and cuboids. Suitable for Grades 6-8.

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