Sure, let's break down the solution step by step:
### SECTION A (25 MARKS)
#### 1. The mass of an empty density bottle is 20 grams. Its mass when filled with water is 40.0 grams and 50.0 grams when filled with liquid X. Given that the density of water is 1000 kg/m³, calculate:
### (a) The volume of the bottle
(2 marks)
First, convert the density of water into grams per cubic centimeter (g/cm³) for consistency with the given masses:
[tex]\[
\text{Density of water} = 1000 \, \text{kg/m}^3 = 1 \, \text{g/cm}^3
\][/tex]
Next, determine the mass of the water in the bottle. This is the difference between the mass of the bottle filled with water and the mass of the empty bottle:
[tex]\[
\text{Mass of water} = \text{Mass of bottle filled with water} - \text{Mass of empty bottle}
\][/tex]
[tex]\[
\text{Mass of water} = 40 \, \text{grams} - 20 \, \text{grams} = 20 \, \text{grams}
\][/tex]
Using the formula for density:
[tex]\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\][/tex]
Solving for the volume of the bottle:
[tex]\[
\text{Volume of the bottle} = \frac{\text{Mass of water}}{\text{Density of water}}
\][/tex]
[tex]\[
\text{Volume of the bottle} = \frac{20 \, \text{grams}}{1 \, \text{g/cm}^3} = 20 \, \text{cm}^3
\][/tex]
### (b) The density of liquid X
(3 marks)
First, determine the mass of liquid X in the bottle. This is the difference between the mass of the bottle filled with liquid X and the mass of the empty bottle:
[tex]\[
\text{Mass of liquid X} = \text{Mass of bottle filled with liquid X} - \text{Mass of empty bottle}
\][/tex]
[tex]\[
\text{Mass of liquid X} = 50 \, \text{grams} - 20 \, \text{grams} = 30 \, \text{grams}
\][/tex]
Using the same density formula and solving for the density of liquid X:
[tex]\[
\text{Density of liquid X} = \frac{\text{Mass of liquid X}}{\text{Volume of the bottle}}
\][/tex]
[tex]\[
\text{Density of liquid X} = \frac{30 \, \text{grams}}{20 \, \text{cm}^3} = 1.5 \, \text{g/cm}^3
\][/tex]
By following these steps, we can clearly see:
- The volume of the bottle is 20 cm³.
- The density of liquid X is 1.5 g/cm³.
These answers allow you to understand both the volume the bottle can contain and compare the density of liquid X with that of water.
