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

Options

A. `0.0855"m"^(3)`
B. `0.086"m"^(3)`
C. `0.08556"m"^(3)`
D. `0.08"m"^(3)`

Answer

To solve the problem, we need to calculate the area and volume of the metal sheet using the given dimensions. We will also ensure to express the final answers in the correct number of significant figures.

### Step 1: Convert all measurements to the same unit
The dimensions given are:
- Length (L) = 4.234 m
- Breadth (B) = 1.005 m
- Thickness (T) = 2.01 cm

First, we need to convert the thickness from centimeters to meters:
\[ T = 2.01 \, \text{cm} = \frac{2.01}{100} \, \text{m} = 0.0201 \, \text{m} \]

### Step 2: Calculate the area
The area of the metal sheet can be calculated using the formula:
\[ \text{Area} = 2(L \times B + B \times T + T \times L) \]

Now substituting the values:
1. Calculate \( L \times B \):
\[ L \times B = 4.234 \, \text{m} \times 1.005 \, \text{m} = 4.26217 \, \text{m}^2 \]

2. Calculate \( B \times T \):
\[ B \times T = 1.005 \, \text{m} \times 0.0201 \, \text{m} = 0.020201 \, \text{m}^2 \]

3. Calculate \( T \times L \):
\[ T \times L = 0.0201 \, \text{m} \times 4.234 \, \text{m} = 0.0850 \, \text{m}^2 \]

Now, summing these areas:
\[ \text{Total Area} = 4.26217 + 0.020201 + 0.0850 = 4.367371 \, \text{m}^2 \]

Finally, multiply by 2:
\[ \text{Area} = 2 \times 4.367371 = 8.734742 \, \text{m}^2 \]

### Step 3: Determine significant figures for area
The least number of significant figures in the measurements is 3 (from 2.01 cm). Therefore, we round the area to 3 significant figures:
\[ \text{Area} \approx 8.73 \, \text{m}^2 \]

### Step 4: Calculate the volume
The volume of the metal sheet can be calculated using the formula:
\[ \text{Volume} = L \times B \times T \]

Substituting the values:
\[ \text{Volume} = 4.234 \, \text{m} \times 1.005 \, \text{m} \times 0.0201 \, \text{m} \]

Calculating:
1. Calculate \( L \times B \):
\[ L \times B = 4.234 \times 1.005 = 4.26217 \, \text{m}^2 \]

2. Now multiply by \( T \):
\[ \text{Volume} = 4.26217 \, \text{m}^2 \times 0.0201 \, \text{m} = 0.0858 \, \text{m}^3 \]

### Step 5: Determine significant figures for volume
The least number of significant figures in the measurements is 3 (from 2.01 cm). Therefore, we round the volume to 3 significant figures:
\[ \text{Volume} \approx 0.0858 \, \text{m}^3 \]

### Final Answers:
- Area = \( 8.73 \, \text{m}^2 \)
- Volume = \( 0.0858 \, \text{m}^3 \)