The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm, and the sum of the lengths of all its edges is 144 cm. The volume, in cubic cm, of the sphere is

The correct option is 1125 π√2.

Let the dimensions (length, width, and height) of the inscribed rectangular box be represented by $x$, $y$, and $z$.

We are given the following information about the rectangular box:
1. The sum of the lengths of all its edges is 144 cm.
2. The total surface area of the box is 846 sq cm.

Step 1: Express the sum of the dimensions
A rectangular box has 12 edges in total (4 lengths, 4 widths, and 4 heights). Therefore, the sum of all its edges is:
$4(x+y+z)=144$
Dividing both sides by 4 gives:
$x+y+z=36$

Step 2: Express the total surface area
The total surface area of a closed rectangular box is given by:
$2(xy+yz+zx)=846$

Step 3: Relate the box dimensions to the sphere's radius
Since the rectangular box is inscribed in a sphere of radius $R$, the space diagonal of the box is equal to the diameter of the sphere ($2R$). Using the 3D Pythagorean theorem, the space diagonal is:
$\sqrt{x^2+y^2+z^2}=2R$
Squaring both sides, we get:
$x^2+y^2+z^2=4R^2$

We can find the value of $x^2+y^2+z^2$ using the algebraic identity:
$(x+y+z{)}^2=x^2+y^2+z^2+2(xy+yz+zx)$
Substituting the known values from Steps 1 and 2:
${36}^2=4R^2+846$
$1296=4R^2+846$
Subtract 846 from both sides:
$4R^2=450$
Solve for $R^2$:
$R^2=\frac{450}{4}=\frac{225}{2}$
Taking the square root of both sides gives the radius of the sphere:
$R=\frac{15}{\sqrt{2}}$

Step 4: Calculate the volume of the sphere
The volume $V$ of a sphere is given by:
$V=\frac{4}{3}\pi R^3$
Substitute the radius $R=\frac{15}{\sqrt{2}}$:
$V=\frac{4}{3}\pi {\left(\frac{15}{\sqrt{2}}\right)}^3$
$V=\frac{4}{3}\pi \left(\frac{3375}{2\sqrt{2}}\right)$
Simplifying the constant terms:
$V=\frac{4\times 3375}{6\sqrt{2}}\pi =\frac{13500}{6\sqrt{2}}\pi =\frac{2250}{\sqrt{2}}$
Multiply the numerator and denominator by $\sqrt{2}$ to rationalize the denominator:
$V=\frac{2250\sqrt{2}}{2}\pi =1125\pi \sqrt{2}$

Therefore, the volume of the sphere is 1125 π√2 cubic cm.