Sure, let's solve step-by-step following the detailed method to find the density of the marble in [tex]\( \text{kg/m}^3 \)[/tex].
### Part (a): Dimensions of a Physical Quantity
The dimension of density involves the fundamental units of mass, length, and time. For density, which is mass per unit volume:
- Mass has the dimension [tex]\( [M] \)[/tex].
- Length has the dimension [tex]\( [L] \)[/tex].
The volume is given by [tex]\( \text{length}^3 \)[/tex], so it has the dimension [tex]\( [L^3] \)[/tex].
Thus, density has the dimension of mass divided by volume. Therefore, the dimensional formula for density [tex]\( \rho \)[/tex] is:
[tex]\[ \left[\rho\right] = \frac{[M]}{[L^3]} = [M L^{-3}] \][/tex]
### Part (b): Definition of Density
Density ([tex]\( \rho \)[/tex]) of a substance is defined as the mass ([tex]\( m \)[/tex]) per unit volume ([tex]\( V \)[/tex]) of the substance. Mathematically, it is expressed as:
[tex]\[ \rho = \frac{m}{V} \][/tex]
### Part (c): Calculation of Density
1. Given values:
- Volume of the marble: [tex]\( 25 \text{ cm}^3 \)[/tex]
- Mass of the marble: [tex]\( 45 \text{ g} \)[/tex]
2. Convert mass from grams to kilograms:
[tex]\[
45 \text{ g} = 45 \times 10^{-3} \text{ kg} = 0.045 \text{ kg}
\][/tex]
3. Convert volume from cubic centimeters to cubic meters:
1 cubic meter ([tex]\( \text{m}^3 \)[/tex]) is [tex]\( 10^6 \)[/tex] cubic centimeters ([tex]\( \text{cm}^3 \)[/tex]). Therefore:
[tex]\[
25 \text{ cm}^3 = 25 \times 10^{-6} \text{ m}^3 = 2.5 \times 10^{-5} \text{ m}^3
\][/tex]
4. Calculate the density in kg/m[tex]\(^3\)[/tex]:
[tex]\[
\rho = \frac{\text{mass}}{\text{volume}} = \frac{0.045 \text{ kg}}{2.5 \times 10^{-5} \text{ m}^3}
\][/tex]
5. Simplify the expression:
[tex]\[
\rho = \frac{0.045}{2.5 \times 10^{-5}} \text{ kg/m}^3
\][/tex]
6. Perform the division:
[tex]\[
\rho = 1800 \text{ kg/m}^3
\][/tex]
So, the density of the marble is [tex]\( 1800 \text{ kg/m}^3 \)[/tex].
