Question

comparing densities
if two balls have the same volume, but ball a has twice as much mass as ball b, which one will have the greater density?
if ball c is 3 times the volume of ball d and ball d has 1/3 the mass of ball c, which has the greater density?
if two balls have the same mass, but ball p is twice as large as ball q, which one will have the greater density?
if ball x is twice as big as ball y and weighs only half as much as ball y, then which one will have the greater density?

Explanation:

To solve these density - comparison problems, we use the formula for density, $
ho=\frac{m}{V}$, where $
ho$ is the density, $m$ is the mass, and $V$ is the volume.

Problem 1:

Step 1: Define the variables

Let the volume of ball A and ball B be $V_A = V_B=V$. Let the mass of ball B be $m_B = m$, then the mass of ball A is $m_A = 2m$.

Step 2: Calculate the densities

The density of ball A, $
ho_A=\frac{m_A}{V_A}=\frac{2m}{V}$. The density of ball B, $
ho_B=\frac{m_B}{V_B}=\frac{m}{V}$. Since $\frac{2m}{V}>\frac{m}{V}$, ball A has a greater density.

Problem 2:

Step 1: Define the variables

Let the volume of ball D be $V_D = V$, then the volume of ball C is $V_C = 3V$. Let the mass of ball C be $m_C = m$, then the mass of ball D is $m_D=\frac{1}{3}m$.

Step 2: Calculate the densities

The density of ball C, $
ho_C=\frac{m_C}{V_C}=\frac{m}{3V}$. The density of ball D, $
ho_D=\frac{m_D}{V_D}=\frac{\frac{1}{3}m}{V}=\frac{m}{3V}$. So, both ball C and ball D have the same density.

Problem 3:

Step 1: Define the variables

Let the mass of ball P and ball Q be $m_P = m_Q=m$. Let the volume of ball Q be $V_Q = V$, then the volume of ball P is $V_P = 2V$.

Step 2: Calculate the densities

The density of ball P, $
ho_P=\frac{m_P}{V_P}=\frac{m}{2V}$. The density of ball Q, $
ho_Q=\frac{m_Q}{V_Q}=\frac{m}{V}$. Since $\frac{m}{V}>\frac{m}{2V}$, ball Q has a greater density.

Problem 4:

Step 1: Define the variables

Let the volume of ball Y be $V_Y = V$, then the volume of ball X is $V_X = 2V$. Let the mass of ball Y be $m_Y = m$, then the mass of ball X is $m_X=\frac{1}{2}m$.

Step 2: Calculate the densities

The density of ball X, $
ho_X=\frac{m_X}{V_X}=\frac{\frac{1}{2}m}{2V}=\frac{m}{4V}$. The density of ball Y, $
ho_Y=\frac{m_Y}{V_Y}=\frac{m}{V}$. Since $\frac{m}{V}>\frac{m}{4V}$, ball Y has a greater density.

Answer:

s:

1. Ball A
2. Both have the same density
3. Ball Q
4. Ball Y