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Sagot :
Sure! Let's go through the problem step by step to calculate the missing values in the table and determine the color and thickness of each layer in the measuring cylinder.
### Part (a): Calculating the missing values in the table
We are given:
- Ethanol with a volume of 20.00 cm³ and density of 0.79 g/cm³
- Glycerin with a mass of 20.00 g and density of 1.26 g/cm³
- Olive oil with a mass of 25.90 g and a volume of 28.80 cm³
- Turpentine with a mass of 30.00 g and a volume of 35.30 cm³
#### For Ethanol
To find the mass of ethanol (i), we use the formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
Thus,
[tex]\[ \text{Mass} = \text{Density} \times \text{Volume} \][/tex]
[tex]\[ \text{Mass}_{\text{ethanol}} = 0.79 \, \text{g/cm}^3 \times 20.00 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Mass}_{\text{ethanol}} = 15.8 \, \text{g} \][/tex]
#### For Glycerin
To find the volume of glycerin (ii), we use the formula:
[tex]\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \][/tex]
[tex]\[ \text{Volume}_{\text{glycerin}} = \frac{20.00 \, \text{g}}{1.26 \, \text{g/cm}^3} \][/tex]
[tex]\[ \text{Volume}_{\text{glycerin}} \approx 15.873 \, \text{cm}^3 \][/tex]
#### For Olive Oil
To find the density of olive oil (iii), we use the formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
[tex]\[ \text{Density}_{\text{olive oil}} = \frac{25.90 \, \text{g}}{28.80 \, \text{cm}^3} \][/tex]
[tex]\[ \text{Density}_{\text{olive oil}} \approx 0.899 \, \text{g/cm}^3 \][/tex]
#### For Turpentine
To find the density of turpentine (iv), we use the same formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
[tex]\[ \text{Density}_{\text{turpentine}} = \frac{30.00 \, \text{g}}{35.30 \, \text{cm}^3} \][/tex]
[tex]\[ \text{Density}_{\text{turpentine}} \approx 0.850 \, \text{g/cm}^3 \][/tex]
So the completed table is:
[tex]\[ \begin{tabular}{|l|l|c|c|c|} \cline { 2 - 5 } \multicolumn{1}{c|}{} & Liquid & Mass/g & \begin{tabular}{c} Volume / \\ \text{cm}^3 \\ \end{tabular} & \begin{tabular}{c} Density / \\ \text{g/cm}^3 \\ \end{tabular} \\ \hline clear & ethanol & 15.8 & 20.00 & 0.79 \\ \hline red & glycerin & 20.00 & 15.873 & 1.26 \\ \hline green & olive oil & 25.90 & 28.80 & 0.899 \\ \hline blue & turpentine & 30.00 & 35.30 & 0.850 \\ \hline \end{tabular} \][/tex]
### Part (b): Determining the color of the liquid in each layer and the thickness of each layer
We need to calculate the thickness of each layer in the 10 cm tall measuring cylinder.
The total volume is 100 cm³. Since volumes sum to the total height, the thickness of each layer is directly proportional to the volume of the liquid:
[tex]\[ \text{Thickness of layer} = \frac{\text{Volume}_{\text{liquid}}}{100 \, \text{cm}^3} \times 10 \, \text{cm} \][/tex]
#### Thickness Calculations
- Ethanol: [tex]\[ \frac{20.00}{100} \times 10 = 2.00 \, \text{cm} \][/tex]
- Glycerin: [tex]\[ \frac{15.873}{100} \times 10 \approx 1.587 \, \text{cm} \][/tex]
- Olive Oil: [tex]\[ \frac{28.80}{100} \times 10 \approx 2.880 \, \text{cm} \][/tex]
- Turpentine: [tex]\[ \frac{35.30}{100} \times 10 \approx 3.530 \, \text{cm} \][/tex]
Next, we sort the liquids by density to determine the order of the layers:
- Ethanol: 0.79 g/cm³ (clear)
- Turpentine: 0.850 g/cm³ (blue)
- Olive Oil: 0.899 g/cm³ (green)
- Glycerin: 1.26 g/cm³ (red)
#### Final Layer Arrangement
[tex]\[ \begin{tabular}{|l|l|l|} \hline & Colour of layer & Thickness of layer (cm) \\ \hline & clear & 2.00 \\ \hline & blue & 3.53 \\ \hline & green & 2.88 \\ \hline & red & 1.587 \\ \hline \end{tabular} \][/tex]
So the colors of the layers from bottom to top would be:
- Ethanol: 2.00 cm (clear)
- Turpentine: 3.53 cm (blue)
- Olive Oil: 2.88 cm (green)
- Glycerin: 1.587 cm (red)
That concludes the step-by-step solution to the problem!
### Part (a): Calculating the missing values in the table
We are given:
- Ethanol with a volume of 20.00 cm³ and density of 0.79 g/cm³
- Glycerin with a mass of 20.00 g and density of 1.26 g/cm³
- Olive oil with a mass of 25.90 g and a volume of 28.80 cm³
- Turpentine with a mass of 30.00 g and a volume of 35.30 cm³
#### For Ethanol
To find the mass of ethanol (i), we use the formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
Thus,
[tex]\[ \text{Mass} = \text{Density} \times \text{Volume} \][/tex]
[tex]\[ \text{Mass}_{\text{ethanol}} = 0.79 \, \text{g/cm}^3 \times 20.00 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Mass}_{\text{ethanol}} = 15.8 \, \text{g} \][/tex]
#### For Glycerin
To find the volume of glycerin (ii), we use the formula:
[tex]\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \][/tex]
[tex]\[ \text{Volume}_{\text{glycerin}} = \frac{20.00 \, \text{g}}{1.26 \, \text{g/cm}^3} \][/tex]
[tex]\[ \text{Volume}_{\text{glycerin}} \approx 15.873 \, \text{cm}^3 \][/tex]
#### For Olive Oil
To find the density of olive oil (iii), we use the formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
[tex]\[ \text{Density}_{\text{olive oil}} = \frac{25.90 \, \text{g}}{28.80 \, \text{cm}^3} \][/tex]
[tex]\[ \text{Density}_{\text{olive oil}} \approx 0.899 \, \text{g/cm}^3 \][/tex]
#### For Turpentine
To find the density of turpentine (iv), we use the same formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
[tex]\[ \text{Density}_{\text{turpentine}} = \frac{30.00 \, \text{g}}{35.30 \, \text{cm}^3} \][/tex]
[tex]\[ \text{Density}_{\text{turpentine}} \approx 0.850 \, \text{g/cm}^3 \][/tex]
So the completed table is:
[tex]\[ \begin{tabular}{|l|l|c|c|c|} \cline { 2 - 5 } \multicolumn{1}{c|}{} & Liquid & Mass/g & \begin{tabular}{c} Volume / \\ \text{cm}^3 \\ \end{tabular} & \begin{tabular}{c} Density / \\ \text{g/cm}^3 \\ \end{tabular} \\ \hline clear & ethanol & 15.8 & 20.00 & 0.79 \\ \hline red & glycerin & 20.00 & 15.873 & 1.26 \\ \hline green & olive oil & 25.90 & 28.80 & 0.899 \\ \hline blue & turpentine & 30.00 & 35.30 & 0.850 \\ \hline \end{tabular} \][/tex]
### Part (b): Determining the color of the liquid in each layer and the thickness of each layer
We need to calculate the thickness of each layer in the 10 cm tall measuring cylinder.
The total volume is 100 cm³. Since volumes sum to the total height, the thickness of each layer is directly proportional to the volume of the liquid:
[tex]\[ \text{Thickness of layer} = \frac{\text{Volume}_{\text{liquid}}}{100 \, \text{cm}^3} \times 10 \, \text{cm} \][/tex]
#### Thickness Calculations
- Ethanol: [tex]\[ \frac{20.00}{100} \times 10 = 2.00 \, \text{cm} \][/tex]
- Glycerin: [tex]\[ \frac{15.873}{100} \times 10 \approx 1.587 \, \text{cm} \][/tex]
- Olive Oil: [tex]\[ \frac{28.80}{100} \times 10 \approx 2.880 \, \text{cm} \][/tex]
- Turpentine: [tex]\[ \frac{35.30}{100} \times 10 \approx 3.530 \, \text{cm} \][/tex]
Next, we sort the liquids by density to determine the order of the layers:
- Ethanol: 0.79 g/cm³ (clear)
- Turpentine: 0.850 g/cm³ (blue)
- Olive Oil: 0.899 g/cm³ (green)
- Glycerin: 1.26 g/cm³ (red)
#### Final Layer Arrangement
[tex]\[ \begin{tabular}{|l|l|l|} \hline & Colour of layer & Thickness of layer (cm) \\ \hline & clear & 2.00 \\ \hline & blue & 3.53 \\ \hline & green & 2.88 \\ \hline & red & 1.587 \\ \hline \end{tabular} \][/tex]
So the colors of the layers from bottom to top would be:
- Ethanol: 2.00 cm (clear)
- Turpentine: 3.53 cm (blue)
- Olive Oil: 2.88 cm (green)
- Glycerin: 1.587 cm (red)
That concludes the step-by-step solution to the problem!
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