Q.6
The length, breadth, and thickness of a strip having a uniform cross-section are measured to be 10.5 cm, 0.05 mm, and 6.0 µm, respectively. Which of the following option(s) give(s) the volume of the strip in cm³ with correct significant figures?

Options:
(A) $3.2\times {10}^{-5}$
(B) $32.0\times {10}^{-6}$
(C) $3.0\times {10}^{-5}$
(D) $3\times {10}^{-5}$

Answer: D

---

Solution:

1. Understanding the Problem

• Given Measurements:

  • Length ($L$) = 10.5 cm (3 significant figures)

  • Breadth ($b$) = 0.05 mm (1 significant figure)

  • Thickness ($t$) = 6.0 µm (2 significant figures)

• Goal: Calculate the volume $V=L\times b\times t$ in cm³ and express it with the correct number of significant figures.

2. Convert All Measurements to cm

First, ensure all dimensions are in the same unit (cm) for consistent volume calculation.

• Length ($L$): Already in cm.

  $$\mathrm{<span\; style="background-color:\; white;">L</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">10.5</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">cm</span>}$$

• Breadth ($b$): Convert mm to cm.

  $$\mathrm{<span\; style="background-color:\; white;">b</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.05</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">mm</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.005</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">cm</span>}$$

• Thickness ($t$): Convert µm to cm.

  $$\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">6.0</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\mathrm{<span\; style="background-color:\; white;">\mu </span>}\text{<span style="background-color: white;">m</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">6.0</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">cm</span>}$$

3. Calculate the Volume

Multiply the three dimensions to find the volume:

$$\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">L</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">b</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">10.5</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">cm</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">0.005</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">cm</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">6.0</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">cm</span>}$$$$\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">10.5</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">0.005</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">6.0</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">cm</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}$$$$\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.315</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">cm</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3.15</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">5</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">cm</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}$$

4. Determine Significant Figures

The number of significant figures in the final result is governed by the measurement with the least number of significant figures:

• Length (10.5 cm): 3 significant figures

• Breadth (0.05 mm = 0.005 cm): 1 significant figure (leading zeros are not significant)

• Thickness (6.0 µm = 6.0 × 10⁻⁴ cm): 2 significant figures

The breadth has the fewest significant figures (1), so the final volume must be rounded to 1 significant figure.

5. Round the Volume

$$\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3.15</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">5</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">cm</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">3</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">5</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">cm</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">rounded to 1 significant figure</span>}<span\; style="background-color:\; white;">)</span>$$

6. Evaluate the Options

• (A) $3.2\times {10}^{-5}$: 2 significant figures (incorrect).

• (B) $32.0\times {10}^{-6}$: 3 significant figures (incorrect).

• (C) $3.0\times {10}^{-5}$: 2 significant figures (incorrect).

• (D) $3\times {10}^{-5}$: 1 significant figure (correct).

7. Final Answer

The correct option is D, as it matches the volume with the proper number of significant figures.

\boxed{D}