Answered

A wooden block with a cuboid shape has a length of 15 cm, a width of [tex]$13000 \mu m$[/tex], and a height of [tex]$14 \times 10^{-4} m$[/tex]. If the mass of the wooden block is [tex][tex]$15 \times 10^{-6} g$[/tex][/tex], calculate the density of the wooden block in SI units.

[Given: [tex]\rho=\frac{m}{v}[/tex]]



Answer :

GinnyAnswer
To calculate the density of the wooden block in SI units, let's follow each step carefully.

### Given:
- Length: [tex]\( l = 15 \text{ cm} \)[/tex]
- Width: [tex]\( w = 13000 \text{ micrometers} (\mu m) \)[/tex]
- Height: [tex]\( h = 14 \times 10^{-4} \text{ m} \)[/tex]
- Mass: [tex]\( m = 15 \times 10^{-6} \text{ g} \)[/tex]

### Step 1: Convert all dimensions to meters (SI unit)
1. Convert length from centimeters to meters:
[tex]\( \text{Length (in meters)} = \frac{15 \text{ cm}}{100} = 0.15 \text{ m} \)[/tex]

2. Convert width from micrometers to meters:
[tex]\( \text{Width (in meters)} = 13000 \times 10^{-6} \text{ m} = 0.013 \text{ m} \)[/tex]

3. Since the height is already given in meters, we use it directly:
[tex]\( \text{Height (in meters)} = 14 \times 10^{-4} \text{ m} = 0.0014 \text{ m} \)[/tex]

### Step 2: Convert mass from grams to kilograms (SI unit)
[tex]\( \text{Mass (in kilograms)} = 15 \times 10^{-6} \text{ g} \times 10^{-3} = 1.5 \times 10^{-11} \text{ kg} \)[/tex]

### Step 3: Calculate the volume of the cuboid
Volume [tex]\( V \)[/tex] is given by:
[tex]\[ V = l \times w \times h \][/tex]
Plugging in the values:
[tex]\[ V = 0.15 \text{ m} \times 0.013 \text{ m} \times 0.0014 \text{ m} = 2.73 \times 10^{-6} \text{ m}^3 \][/tex]

### Step 4: Calculate the density
Density [tex]\( \rho \)[/tex] is given by:
[tex]\[ \rho = \frac{m}{V} \][/tex]
Plugging in the mass ([tex]\( m \)[/tex]) and volume ([tex]\( V \)[/tex]):
[tex]\[ \rho = \frac{1.5 \times 10^{-11} \text{ kg}}{2.73 \times 10^{-6} \text{ m}^3} = 5.4945 \times 10^{-6} \text{ kg/m}^3 \][/tex]

### Final Answer:
The density of the wooden block in SI unit is [tex]\( \rho = 5.4945 \times 10^{-6} \text{ kg/m}^3 \)[/tex].