Sure! Let's solve the problem step-by-step.
### (i) Calculate the volume of the brick before holes are made.
The volume [tex]\( V \)[/tex] of a rectangular prism can be calculated using the formula:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Given dimensions:
- Length = 21 cm
- Width = 10 cm
- Height = 8 cm
Substituting the given values:
[tex]\[ V = 21 \, \text{cm} \times 10 \, \text{cm} \times 8 \, \text{cm} \][/tex]
[tex]\[ V = 1680 \, \text{cm}^3 \][/tex]
So, the volume of the brick before holes are made is
[tex]\[ \boxed{1680 \, \text{cm}^3} \][/tex]
### (ii) Calculate the volume of the two holes.
The volume [tex]\( V_{\text{cylinder}} \)[/tex] of a cylinder can be calculated using the formula:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
Given:
- Diameter of each hole = 3 cm
- Therefore, the radius [tex]\( r \)[/tex] of each hole = [tex]\( \frac{3}{2} = 1.5 \, \text{cm} \)[/tex]
- Height [tex]\( h \)[/tex] of each hole = 8 cm (same as the height of the brick)
Using [tex]\( \pi \approx 3.14 \)[/tex]:
Volume of one hole:
[tex]\[ V_{\text{hole}} = 3.14 \times (1.5 \, \text{cm})^2 \times 8 \, \text{cm} \][/tex]
[tex]\[ V_{\text{hole}} = 3.14 \times 2.25 \, \text{cm}^2 \times 8 \, \text{cm} \][/tex]
[tex]\[ V_{\text{hole}} = 3.14 \times 18 \, \text{cm}^3 \][/tex]
[tex]\[ V_{\text{hole}} \approx 56.52 \, \text{cm}^3 \][/tex]
Since there are two holes, the total volume of the two holes is:
[tex]\[ V_{\text{total\_holes}} = 2 \times 56.52 \, \text{cm}^3 \][/tex]
[tex]\[ V_{\text{total\_holes}} \approx 113.04 \, \text{cm}^3 \][/tex]
So, the volume of the two holes is approximately
[tex]\[ \boxed{113.04 \, \text{cm}^3} \][/tex]
### (iii) Calculate the volume of clay that the brick is made of.
The volume of clay [tex]\( V_{\text{clay}} \)[/tex] can be found by subtracting the total volume of the holes from the volume of the brick.
[tex]\[ V_{\text{clay}} = V_{\text{brick}} - V_{\text{total\_holes}} \][/tex]
[tex]\[ V_{\text{clay}} = 1680 \, \text{cm}^3 - 113.04 \, \text{cm}^3 \][/tex]
[tex]\[ V_{\text{clay}} \approx 1566.96 \, \text{cm}^3 \][/tex]
So, the volume of clay that the brick is made of is approximately
[tex]\[ \boxed{1566.96 \, \text{cm}^3} \][/tex]
