the density of a material is 8 g/cc. In a system in which units of length is 5 cm and of mass is 20 g, the density of material is :

Options

A. 50
B. 40
C. 80
D. 24

Answer

To solve the problem, we need to find the density of a material in a new system of units where the unit of length is 5 cm and the unit of mass is 20 g. The given density of the material is 8 g/cm³.

### Step-by-Step Solution:

1. **Understand the formula for density**:
Density (ρ) is defined as mass (m) divided by volume (V).
\[ \rho = \frac{m}{V} \]

2. **Identify the given values**:
- Density in the first system (ρ₁) = 8 g/cm³
- Mass in the second system (m₂) = 20 g
- Length in the second system (l₂) = 5 cm

3. **Calculate the volume in the second system**:
Since the unit of length is 5 cm, the volume (V₂) in the second system can be calculated as:
\[ V = \text{length}^3 = (5 \text{ cm})^3 = 125 \text{ cm}^3 \]

4. **Calculate the density in the second system**:
Using the formula for density, we can substitute the values:
\[ \rho_2 = \frac{m_2}{V_2} = \frac{20 \text{ g}}{125 \text{ cm}^3} \]

5. **Simplify the expression**:
Now, we simplify the fraction:
\[ \rho_2 = \frac{20}{125} \text{ g/cm}^3 \]
This can be simplified further:
\[ \rho_2 = \frac{20 \div 5}{125 \div 5} = \frac{4}{25} \text{ g/cm}^3 \]

6. **Convert the density to the same units as the first system**:
We know that the density in the first system is 8 g/cm³. To find the numerical value in the second system, we can use the relationship between the two systems:
\[ \rho_2 = \rho_1 \times \frac{V_1}{V_2} \]
Here, the volume in the first system is derived from the density:
\[ V_1 = \frac{m_1}{\rho_1} = \frac{20 g}{8 g/cm^3} = 2.5 cm^3 \]

7. **Calculate the new density**:
Now we can substitute the values:
\[ \rho_2 = 8 \times \frac{125}{20} = 8 \times 6.25 = 50 \]

Thus, the density of the material in the new system is **50 g/cm³**.