To determine the density of each pair of balls, we can use the formula for density, which is given by:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
1. Comparing Ball A and Ball B:
- Let the volume of both balls be \( V \).
- Let the mass of Ball A be \( M_A = 2M_B \) (since Ball A has twice the mass of Ball B).
The densities are:
- For Ball A: \( \text{Density}_A = \frac{M_A}{V} = \frac{2M_B}{V} \)
- For Ball B: \( \text{Density}_B = \frac{M_B}{V} \)
Since \( \text{Density}_A = \frac{2M_B}{V} \) and \( \text{Density}_B = \frac{M_B}{V} \), it follows that:
\[ \text{Density}_A = 2 \times \text{Density}_B \]
Conclusion: Ball A has the greater density.
2. Comparing Ball C and Ball D:
- Let the volume of Ball D be \( V_D \). Then, Ball C's volume is \( V_C = 3V_D \).
- Let the mass of Ball C be \( M_C \) and the mass of Ball D be \( M_D = \frac{1}{3}M_C \).
The densities are:
- For Ball C: \( \text{Density}_C = \frac{M_C}{3V_D} \)
- For Ball D: \( \text{Density}_D = \frac{M_D}{V_D} = \frac{\frac{1}{3}M_C}{V_D} = \frac{M_C}{3V_D} \)
Since \( \text{Density}_C = \text{Density}_D \):
Conclusion: Balls C and D have the same density.
3. Comparing Ball P and Ball Q:
- Let the mass of both balls be \( M \).
- Let the volume of Ball Q be \( V_Q \). Then, volume of Ball P is \( V_P = 8V_Q \) (since Ball P is twice as large in each dimension, volume scales as \( 2^3 = 8 \)).
The densities are:
- For Ball P: \( \text{Density}_P = \frac{M}{8V_Q} \)
- For Ball Q: \( \text{Density}_Q = \frac{M}{V_Q} \)
Since \( \text{Density}_P = \frac{M}{8V_Q} \) and \( \text{Density}_Q = \frac{M}{V_Q} \):
\[ \text{Density}_P = \frac{1}{8} \times \text{Density}_Q \]
Conclusion: Ball Q has the greater density.
4. Comparing Ball X and Ball Y:
- Let the volume of Ball Y be \( V_Y \). Then, volume of Ball X is \( V_X = 8V_Y \) (as Ball X is twice as big in each dimension).
- Let the mass of Ball Y be \( M_Y \). Then, the mass of Ball X is \( M_X = \frac{1}{2}M_Y \) (since Ball X weighs only half as much as Ball Y).
The densities are:
- For Ball X: \( \text{Density}_X = \frac{M_X}{8V_Y} = \frac{\frac{1}{2}M_Y}{8V_Y} = \frac{M_Y}{16V_Y} \)
- For Ball Y: \( \text{Density}_Y = \frac{M_Y}{V_Y} \)
Comparing the two:
\[ \text{Density}_X = \frac{1}{16} \times \text{Density}_Y \]
Conclusion: Ball Y has the greater density.
To summarize:
- Ball A is denser than Ball B.
- Balls C and D have the same density.
- Ball Q is denser than Ball P.
- Ball Y is denser than Ball X.