The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m and 2.01 cm, respectively. Give the area and volume of the sheet to correct significant figures.

The correct option is 8.72 m² and 0.0855 m³.

Let's solve the problem step-by-step to determine the area and volume of the metal sheet with the correct number of significant figures.

1. Identify the given dimensions:
Length, $l=4.234\text{m}$ (4 significant figures)
Breadth, $b=1.005\text{m}$ (4 significant figures)
Thickness, $t=2.01\text{cm}=2.01\times {10}^{-2}\text{m}=0.0201\text{m}$ (3 significant figures)

2. Calculate the Total Surface Area:
The total surface area $A$ of a rectangular sheet is given by the formula:
$A=2(lb+bt+tl)$

Let's calculate the individual term products:
$lb=4.234\times 1.005=4.25517{\text{m}}^2$
$bt=1.005\times 0.0201=0.0202005{\text{m}}^2$
$tl=0.0201\times 4.234=0.0851034{\text{m}}^2$

Summing these values:
$lb+bt+tl=4.25517+0.0202005+0.0851034=4.3604739{\text{m}}^2$

Multiplying by 2:
$A=2\times 4.3604739=8.7209478{\text{m}}^2$

According to the rules of significant figures, the final result of addition/multiplication should match the least precise measurement. The thickness $t=2.01\text{cm}$ has the least number of significant figures (3 significant figures). Therefore, the total area should be rounded off to 3 significant figures:
$A\approx 8.72{\text{m}}^2$

3. Calculate the Volume:
The volume $V$ of the rectangular sheet is given by:
$V=l\times b\times t$
$V=4.234\times 1.005\times 0.0201=0.085528917{\text{m}}^3$

Since the minimum number of significant figures in the given values is 3 (from the thickness, $2.01\text{cm}$), we round the volume to 3 significant figures:
$V\approx 0.0855{\text{m}}^3$

Thus, the area and volume of the sheet to the correct significant figures are 8.72 m² and 0.0855 m³, respectively.