To calculate the density of the sand, we need its mass and its volume.
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Here's the solution to 9(a):
9. (a)
To calculate the density of the sand, we need its mass and its volume.
Step 1: Determine the mass of the water that fills the bottle.
Mass of water = (Mass of bottle + water) - (Mass of empty bottle)
Mass of water = $98 \text{ g} - 35 \text{ g} = 63 \text{ g}$
Step 2: Determine the volume of the bottle.
Assuming the density of water is $1 \text{ g/cm}^3$.
Volume of bottle = Volume of water that fills the bottle
$$ V_{bottle} = \frac{\text{Mass of water}}{\text{Density of water}} = \frac{63 \text{ g}}{1 \text{ g/cm}^3} = 63 \text{ cm}^3 $$
Step 3: Determine the mass of the sand.
Mass of sand = (Mass of bottle + some sand) - (Mass of empty bottle)
Mass of sand = $65 \text{ g} - 35 \text{ g} = 30 \text{ g}$
Step 4: Determine the volume of the sand.
The line "Mass of bottle + sand + water on top of the sand = 80g" tells us the total mass when the bottle is filled with sand and then water is added to fill the remaining space.
Mass of (sand + water) in the bottle = (Mass of bottle + sand + water) - (Mass of empty bottle)
Mass of (sand + water) = $80 \text{ g} - 35 \text{ g} = 45 \text{ g}$
Now, subtract the mass of the sand (from Step 3) to find the mass of the water added on top of the sand.
Mass of water added = Mass of (sand + water) - Mass of sand
Mass of water added = $45 \text{ g} - 30 \text{ g} = 15 \text{ g}$
Using the density of water ($1 \text{ g/cm}^3$), find the volume of the water added.
Volume of water added = $\frac{\text{Mass of water added}}{\text{Density of water}} = \frac{15 \text{ g}}{1 \text{ g/cm}^3} = 15 \text{ cm}^3$
The volume of the sand is the total volume of the bottle minus the volume of the water added.
Volume of sand = Volume of bottle - Volume of water added
Volume of sand = $63 \text{ cm}^3 - 15 \text{ cm}^3 = 48 \text{ cm}^3$
Step 5: Calculate the density of the sand.
Density of sand = $\frac{\text{Mass of sand}}{\text{Volume of sand}}$
$$ \text{Density of sand} = \frac{30 \text{ g}}{48 \text{ cm}^3} $$
$$ \text{Density of sand} = \frac{5}{8} \text{ g/cm}^3 $$
$$ \text{Density of sand} = 0.625 \text{ g/cm}^3 $$
The density of the sand is $\boxed{0.625 \text{ g/cm}^3}$.
