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Sagot :
Sure, let's work through each part of the problem step-by-step.
### (i) Calculating the Volume of the Brick Before Holes are Made
The brick is in the shape of a rectangular prism. To find the volume of a rectangular prism, we use the formula:
[tex]\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \][/tex]
Given:
- Length [tex]\( = 21 \)[/tex] cm
- Width [tex]\( = 10 \)[/tex] cm
- Height [tex]\( = 8 \)[/tex] cm
[tex]\[ \text{Volume}_{\text{brick}} = 21 \, \text{cm} \times 10 \, \text{cm} \times 8 \, \text{cm} \][/tex]
[tex]\[ \text{Volume}_{\text{brick}} = 1680 \, \text{cm}^3 \][/tex]
Therefore, the volume of the brick before the holes are made is [tex]\( 1680 \, \text{cm}^3 \)[/tex].
### (ii) Calculating the Volume of the Two Holes
Each hole is cylindrical in shape. The formula for the volume of a cylinder is:
[tex]\[ \text{Volume} = \pi r^2 h \][/tex]
Given:
- Diameter of the hole [tex]\( = 3 \)[/tex] cm, so the radius [tex]\( r = \frac{3}{2} = 1.5 \)[/tex] cm
- Height [tex]\( h = 8 \)[/tex] cm (same as the height of the brick)
- [tex]\( \pi \approx 3.14 \)[/tex]
First, calculate the volume of one hole:
[tex]\[ \text{Volume}_{\text{hole}} = 3.14 \times (1.5 \, \text{cm})^2 \times 8 \, \text{cm} \][/tex]
[tex]\[ \text{Volume}_{\text{hole}} = 3.14 \times 2.25 \, \text{cm}^2 \times 8 \, \text{cm} \][/tex]
[tex]\[ \text{Volume}_{\text{hole}} = 3.14 \times 18 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Volume}_{\text{hole}} = 56.52 \, \text{cm}^3 \][/tex]
Since there are two holes, the total volume of both holes is:
[tex]\[ \text{Volume}_{\text{holes total}} = 2 \times 56.52 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Volume}_{\text{holes total}} = 113.04 \, \text{cm}^3 \][/tex]
Therefore, the total volume of the two holes is [tex]\( 113.04 \, \text{cm}^3 \)[/tex].
### (iii) Calculating the Volume of Clay that the Brick is Made Of
To find the volume of the clay, we subtract the total volume of the holes from the volume of the brick before the holes were made.
[tex]\[ \text{Volume}_{\text{clay}} = \text{Volume}_{\text{brick}} - \text{Volume}_{\text{holes total}} \][/tex]
Given:
- [tex]\( \text{Volume}_{\text{brick}} = 1680 \, \text{cm}^3 \)[/tex]
- [tex]\( \text{Volume}_{\text{holes total}} = 113.04 \, \text{cm}^3 \)[/tex]
[tex]\[ \text{Volume}_{\text{clay}} = 1680 \, \text{cm}^3 - 113.04 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Volume}_{\text{clay}} = 1566.96 \, \text{cm}^3 \][/tex]
Therefore, the volume of clay that the brick is made of is [tex]\( 1566.96 \, \text{cm}^3 \)[/tex].
### (i) Calculating the Volume of the Brick Before Holes are Made
The brick is in the shape of a rectangular prism. To find the volume of a rectangular prism, we use the formula:
[tex]\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \][/tex]
Given:
- Length [tex]\( = 21 \)[/tex] cm
- Width [tex]\( = 10 \)[/tex] cm
- Height [tex]\( = 8 \)[/tex] cm
[tex]\[ \text{Volume}_{\text{brick}} = 21 \, \text{cm} \times 10 \, \text{cm} \times 8 \, \text{cm} \][/tex]
[tex]\[ \text{Volume}_{\text{brick}} = 1680 \, \text{cm}^3 \][/tex]
Therefore, the volume of the brick before the holes are made is [tex]\( 1680 \, \text{cm}^3 \)[/tex].
### (ii) Calculating the Volume of the Two Holes
Each hole is cylindrical in shape. The formula for the volume of a cylinder is:
[tex]\[ \text{Volume} = \pi r^2 h \][/tex]
Given:
- Diameter of the hole [tex]\( = 3 \)[/tex] cm, so the radius [tex]\( r = \frac{3}{2} = 1.5 \)[/tex] cm
- Height [tex]\( h = 8 \)[/tex] cm (same as the height of the brick)
- [tex]\( \pi \approx 3.14 \)[/tex]
First, calculate the volume of one hole:
[tex]\[ \text{Volume}_{\text{hole}} = 3.14 \times (1.5 \, \text{cm})^2 \times 8 \, \text{cm} \][/tex]
[tex]\[ \text{Volume}_{\text{hole}} = 3.14 \times 2.25 \, \text{cm}^2 \times 8 \, \text{cm} \][/tex]
[tex]\[ \text{Volume}_{\text{hole}} = 3.14 \times 18 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Volume}_{\text{hole}} = 56.52 \, \text{cm}^3 \][/tex]
Since there are two holes, the total volume of both holes is:
[tex]\[ \text{Volume}_{\text{holes total}} = 2 \times 56.52 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Volume}_{\text{holes total}} = 113.04 \, \text{cm}^3 \][/tex]
Therefore, the total volume of the two holes is [tex]\( 113.04 \, \text{cm}^3 \)[/tex].
### (iii) Calculating the Volume of Clay that the Brick is Made Of
To find the volume of the clay, we subtract the total volume of the holes from the volume of the brick before the holes were made.
[tex]\[ \text{Volume}_{\text{clay}} = \text{Volume}_{\text{brick}} - \text{Volume}_{\text{holes total}} \][/tex]
Given:
- [tex]\( \text{Volume}_{\text{brick}} = 1680 \, \text{cm}^3 \)[/tex]
- [tex]\( \text{Volume}_{\text{holes total}} = 113.04 \, \text{cm}^3 \)[/tex]
[tex]\[ \text{Volume}_{\text{clay}} = 1680 \, \text{cm}^3 - 113.04 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Volume}_{\text{clay}} = 1566.96 \, \text{cm}^3 \][/tex]
Therefore, the volume of clay that the brick is made of is [tex]\( 1566.96 \, \text{cm}^3 \)[/tex].
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