Certainly! Let's go through the step-by-step solution to determine the density of the solid.
### Step 1: Determine the weight of the liquid displaced
Since the solid weighs 277.5 g in air and 212.5 g when immersed in the liquid, we can calculate the weight of the liquid displaced by the buoyant force. The difference in the weights gives us the weight of the displaced liquid.
[tex]\[ \text{Weight of liquid displaced} = \text{Weight in air} - \text{Weight in liquid} \][/tex]
[tex]\[ \text{Weight of liquid displaced} = 277.5 \text{ g} - 212.5 \text{ g} \][/tex]
[tex]\[ \text{Weight of liquid displaced} = 65.0 \text{ g} \][/tex]
### Step 2: Calculate the volume of the liquid displaced
To find the volume of the liquid displaced, we use the density of the liquid (given as 0.9 g/cm³). Since density is mass per unit volume, we can rearrange the formula to find volume:
[tex]\[ \text{Density} (\rho) = \frac{\text{Mass}}{\text{Volume}} \][/tex]
[tex]\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \][/tex]
Using the weight of the displaced liquid:
[tex]\[ \text{Volume of liquid displaced} = \frac{65.0 \text{ g}}{0.9 \text{ g/cm}^3} \][/tex]
[tex]\[ \text{Volume of liquid displaced} = 72.222 \text{ cm}^3 \][/tex]
### Step 3: Calculate the density of the solid
Now that we know the volume of the liquid displaced is equal to the volume of the solid (since the solid is fully immersed), we can calculate the density of the solid. The density of the solid is calculated as its weight in air divided by the volume it displaces.
[tex]\[ \text{Density of solid} = \frac{\text{Weight of solid in air}}{\text{Volume of liquid displaced}} \][/tex]
[tex]\[ \text{Density of solid} = \frac{277.5 \text{ g}}{72.222 \text{ cm}^3} \][/tex]
[tex]\[ \text{Density of solid} \approx 3.84 \text{ g/cm}^3 \][/tex]
Thus, the density of the solid is approximately [tex]\(3.84 \text{ g/cm}^3\)[/tex].
