The edges of a rectangular box are in the ratio of 3 : 5 : 8, and its surface area is 632 cm². What will be the volume of the box?

The correct option is 960 cm³.

Step-by-step Explanation:

Let the dimensions (length, breadth, and height) of the rectangular box be $l$, $b$, and $h$ respectively.

Given that the edges are in the ratio of 3 : 5 : 8, we can represent them in terms of a constant ratio factor $x$:

$l=3x$

$b=5x$

$h=8x$

1. Formula for the Total Surface Area of a Rectangular Box:

$\text{Total Surface Area}=2(lb+bh+hl)$

Given that the total surface area is 632 cm²:

$2\left(\right(3x\left)\right(5x)+(5x\left)\right(8x)+(8x\left)\right(3x\left)\right)=632$

2. Solve for $x$:

$2(15x^2+40x^2+24x^2)=632$

$2\left(79x^2\right)=632$

$158x^2=632$

$x^2=\frac{632}{158}$

$x^2=4$

$x=2$

3. Calculate the Dimensions of the Box:

Length ($l$) = $3\times 2=6\text{cm}$

Breadth ($b$) = $5\times 2=10\text{cm}$

Height ($h$) = $8\times 2=16\text{cm}$

4. Calculate the Volume of the Box:

$\text{Volume}=l\times b\times h$

$\text{Volume}=6\times 10\times 16=960{\text{cm}}^3$

Thus, the volume of the box is 960 cm³.