Problem Analysis:

The problem involves a cuboid where:

1. Length ($l$) is twice the breadth ($b$): $l = 2b$,
2. Height ($h$) is 10 cm: $h = 10$,
3. Total Surface Area (TSA): $736 \, \text{cm}^2$,
4. We are required to find the Volume of the cuboid.

Step-by-Step Solution:

1. Formula for Total Surface Area:

The TSA of a cuboid is given by: $\text{TSA} = 2 \left( lb + bh + hl \right)$

Substitute $l = 2b$ and $h = 10$ into the formula: $\text{TSA} = 2 \left( (2b \cdot b) + (b \cdot 10) + (10 \cdot 2b) \right)$ $\text{TSA} = 2 \left( 2b^2 + 10b + 20b \right)$ $\text{TSA} = 2 \left( 2b^2 + 30b \right)$ $\text{TSA} = 4b^2 + 60b$

Set $\text{TSA} = 736$: $4b^2 + 60b = 736$

2. Simplify the Quadratic Equation:

Divide the equation by 4 to simplify: $b^2 + 15b - 184 = 0$

3. Solve the Quadratic Equation:

Using the quadratic formula: $b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}, \quad \text{where } A = 1, B = 15, C = -184.$

Substitute the values: $b = \frac{-15 \pm \sqrt{15^2 - 4(1)(-184)}}{2(1)}$ $b = \frac{-15 \pm \sqrt{225 + 736}}{2}$ $b = \frac{-15 \pm \sqrt{961}}{2}$ $b = \frac{-15 \pm 31}{2}$

Choose the positive root: $b = \frac{-15 + 31}{2} = \frac{16}{2} = 8 \, \text{cm}.$

4. Calculate Length:

Since $l = 2b$: $l = 2 \cdot 8 = 16 \, \text{cm}.$

5. Calculate Volume:

The Volume ($V$) of a cuboid is given by: $V = l \cdot b \cdot h$ $V = 16 \cdot 8 \cdot 10 = 1280 \, \text{cm}^3.$

Final Answer:

The volume of the cuboid is: $\boxed{1280 \, \text{cm}^3}.$

Let me know if you'd like further details or clarifications! 😊

Related Questions:

1. How is the total surface area formula derived for a cuboid?
2. What is the general method for solving quadratic equations?
3. How would the solution change if the height were doubled?
4. Can we verify the calculations using unit consistency?
5. What happens to the volume if all dimensions are doubled?

Tip:

Always cross-check dimensions (units) when solving geometry problems to ensure consistency!