Each side of a cube is measured to be 5.402 cm. The total surface area and the volume of the cube in appropriate significant figures are :

The correct option is 175.1cm² , 157.6cm³.

Let's break down the step-by-step calculation and understand how to apply the rules of significant figures to find the total surface area and volume of the cube.

1. Understanding the Given Data:
The measured length of each side of the cube, denoted by $a$, is:
$a=5.402\text{ cm}$
This measurement has 4 significant figures (5, 4, 0, and 2).

2. Calculating the Total Surface Area:
The formula for the total surface area ($A$) of a cube is:
$A=6a^2$
Substituting the value of $a$:
$A=6\times {\left(5.402\text{ cm}\right)}^2$
$A=6\times 29.181604{\text{ cm}}^2$
$A=175.089624{\text{ cm}}^2$
According to the rules of multiplication for significant figures, the final result of a calculation must not have more significant figures than the original measurement with the least number of significant figures. Here, the side length $5.402\text{ cm}$ has 4 significant figures (the number 6 is an exact number/constant and does not limit the significant figures). Therefore, we round off the surface area to 4 significant figures:
$A\approx 175.1{\text{ cm}}^2$

3. Calculating the Volume:
The formula for the volume ($V$) of a cube is:
$V=a^3$
Substituting the value of $a$:
$V={\left(5.402\text{ cm}\right)}^3$
$V=157.6390248{\text{ cm}}^3$
Applying the significant figure rule for multiplication, our final volume must also be rounded to 4 significant figures since the side length measurement has 4 significant figures:
$V\approx 157.6{\text{ cm}}^3$

Thus, in appropriate significant figures, the total surface area is 175.1 cm² and the volume is 157.6 cm³.