Length, breadth and thickness of a strip having a uniform cross section are measured to be $10.5\text{cm}$, $0.05\text{mm}$, and $6.0\text{μm}$, respectively.

Which of the following option(s) give(s) the volume of the strip in ${\text{cm}}^3$ with correct significant figures:

The correct answer is:

$3\times {10}^{-5}{\text{cm}}^3$

Step 1: Understand the Given Dimensions
We are given the length ($l$), breadth ($b$), and thickness ($t$) of a strip with a uniform cross-section:
• Length, $l=10.5\text{cm}$ (which has 3 significant figures)
• Breadth, $b=0.05\text{mm}$ (which has 1 significant figure)
• Thickness, $t=6.0\text{μm}$ (which has 2 significant figures)

Step 2: Convert All Units to Centimeters ($\text{cm}$)
Since the volume is required in ${\text{cm}}^3$, we convert all dimensions to centimeters:
• Length is already in $\text{cm}$:
$l=10.5\text{cm}$
• Convert breadth from $\text{mm}$ to $\text{cm}$ (since $1\text{mm}={10}^{-1}\text{cm}$):
$b=0.05\text{mm}=0.05\times {10}^{-1}\text{cm}=0.005\text{cm}=5\times {10}^{-3}\text{cm}$
• Convert thickness from $\text{μm}$ to $\text{cm}$ (since $1\text{μm}={10}^{-6}\text{m}={10}^{-4}\text{cm}$):
$t=6.0\text{μm}=6.0\times {10}^{-4}\text{cm}$

Step 3: Calculate the Volume
The volume $V$ of a rectangular strip is given by the product of its length, breadth, and thickness:
$V=l\times b\times t$

Substituting the converted values into the formula:
$V=10.5\text{cm}\times \left(5\times {10}^{-3}\text{cm}\right)\times \left(6.0\times {10}^{-4}\text{cm}\right)$

Multiply the numerical factors:
$V=\left(10.5\times 5\times 6.0\right)\times {10}^{-7}{\text{cm}}^3$
$V=315\times {10}^{-7}{\text{cm}}^3$
$V=3.15\times {10}^{-5}{\text{cm}}^3$

Step 4: Apply the Rules of Significant Figures for Multiplication
When multiplying numbers, the final result must be rounded to the same number of significant figures as the measurement with the least number of significant figures.
• Length ($10.5\text{cm}$) has 3 significant figures.
• Breadth ($0.05\text{mm}$ or $0.005\text{cm}$) has only 1 significant figure (leading zeros are not significant).
• Thickness ($6.0\text{μm}$) has 2 significant figures.

The minimum number of significant figures among the measurements is 1 (from the breadth). Therefore, the final volume must be rounded to 1 significant figure:

$V\approx 3\times {10}^{-5}{\text{cm}}^3$