To determine which object is the most dense among the four given objects, we need to understand how density is calculated. Density is defined as mass per unit volume and can be represented by the formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
Given that all objects have the same mass, the object with the smallest volume will have the highest density because the same mass is distributed over a smaller space.
Let's go through the volumes of each object step-by-step:
1. Object 1: Volume = [tex]\(6 \, \text{cm}^3\)[/tex]
2. Object 2: Volume = [tex]\(8 \, \text{cm}^3\)[/tex]
3. Object 3: Volume = [tex]\(3 \, \text{cm}^3\)[/tex]
4. Object 4: Volume = [tex]\(14 \, \text{cm}^3\)[/tex]
Since the mass is constant across all objects, we can compare the densities by calculating the reciprocal of the volume for each object (as density is inversely proportional to volume in this case):
1. Density of Object 1:
[tex]\[ \frac{1}{6 \, \text{cm}^3} \approx 0.1667 \, \text{cm}^{-3} \][/tex]
2. Density of Object 2:
[tex]\[ \frac{1}{8 \, \text{cm}^3} = 0.125 \, \text{cm}^{-3} \][/tex]
3. Density of Object 3:
[tex]\[ \frac{1}{3 \, \text{cm}^3} \approx 0.3333 \, \text{cm}^{-3} \][/tex]
4. Density of Object 4:
[tex]\[ \frac{1}{14 \, \text{cm}^3} \approx 0.0714 \, \text{cm}^{-3} \][/tex]
Now, let's compare these densities:
- Density of Object 1: [tex]\(0.1667 \, \text{cm}^{-3}\)[/tex]
- Density of Object 2: [tex]\(0.125 \, \text{cm}^{-3}\)[/tex]
- Density of Object 3: [tex]\(0.3333 \, \text{cm}^{-3}\)[/tex]
- Density of Object 4: [tex]\(0.0714 \, \text{cm}^{-3}\)[/tex]
Of these, [tex]\(0.3333 \, \text{cm}^{-3}\)[/tex] is the highest density.
Therefore, Object 3 is the most dense.
