A cube has a side length of 4 inches and a mass of [tex]$5 \text{ lbs}[tex]$[/tex]. Use the conversion rates below to find the density of the cube in [tex]$[/tex]kg/m^3$[/tex].

[tex]
\begin{array}{l}
1 \text{ inch} = 25 \text{ mm} \\
1 \text{ lb} = 450 \text{ g}
\end{array}
[/tex]



Answer :

Certainly! Let's solve this problem step-by-step.

We have a cube with a side length of 4 inches and a mass of 5 pounds. We need to find the density of the cube in \(\text{kg/m}^3\).

### Step 1: Convert the side length from inches to meters

First, we convert the side length from inches to millimeters:
[tex]\[ 4 \text{ inches} \times 25 \frac{\text{mm}}{\text{inch}} = 100 \text{ mm} \][/tex]

Next, we convert millimeters to meters by knowing that there are 1,000 millimeters in a meter:
[tex]\[ 100 \text{ mm} \times \frac{1 \text{ m}}{1000 \text{ mm}} = 0.1 \text{ m} \][/tex]

So, the side length of the cube is \(0.1 \text{ meters}\).

### Step 2: Convert the mass from pounds to kilograms

We know that:
[tex]\[ 1 \text{ lb} = 450 \text{ g} \][/tex]

Thus, converting 5 pounds to grams:
[tex]\[ 5 \text{ lbs} \times 450 \frac{\text{g}}{\text{lb}} = 2250 \text{ g} \][/tex]

Next, convert grams to kilograms by noting that there are 1,000 grams in a kilogram:
[tex]\[ 2250 \text{ g} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 2.25 \text{ kg} \][/tex]

So, the mass of the cube is \(2.25 \text{ kilograms}\).

### Step 3: Calculate the volume of the cube in cubic meters

The volume \(V\) of a cube is found using the formula \(V = \text{side}^3\):

[tex]\[ V = (0.1 \text{ m})^3 = 0.001 \text{ m}^3 \][/tex]

### Step 4: Calculate the density in \(\text{kg/m}^3\)

Density \(\rho\) is defined as mass (\(m\)) per unit volume (\(V\)):

[tex]\[ \rho = \frac{m}{V} \][/tex]

Substituting the values we obtained:
[tex]\[ \rho = \frac{2.25 \text{ kg}}{0.001 \text{ m}^3} = 2250 \text{ kg/m}^3 \][/tex]

Therefore, the density of the cube is [tex]\(2250 \text{ kg/m}^3\)[/tex].