Let's determine the volume of soil Amira needs to fill the pyramid-shaped plant pot to half of its height.
Step 1: Identify the shape and given dimensions.
- The pot has a square base.
- Side length of the base = 30 cm.
- Height of the pot = 30 cm.
Step 2: Understand the formula for the volume of a pyramid.
The volume [tex]\( V \)[/tex] of a pyramid with a square base can be calculated using the formula:
[tex]\[ V = \frac{1}{3} \times \text{base area} \times \text{height} \][/tex]
Step 3: Calculate the base area.
Since the base is a square:
[tex]\[ \text{Base area} = \text{side length} \times \text{side length} = 30 \, \text{cm} \times 30 \, \text{cm} = 900 \, \text{cm}^2 \][/tex]
Step 4: Calculate the volume when the pyramid is filled to its full height.
Substitute the height (30 cm) and the base area (900 cm²) into the volume formula:
[tex]\[ V_{\text{full}} = \frac{1}{3} \times 900 \, \text{cm}^2 \times 30 \, \text{cm} = 9000 \, \text{cm}^3 \][/tex]
Step 5: Determine the fill height to which Amira wants to fill the pot.
Amira wants to fill the pot to [tex]\(\frac{1}{2}\)[/tex] of its height, so:
[tex]\[ \text{Fill height} = \frac{1}{2} \times 30 \, \text{cm} = 15 \, \text{cm} \][/tex]
Step 6: Calculate the volume when the pyramid is filled to half its height.
Substitute the fill height (15 cm) and the base area (900 cm²) into the volume formula:
[tex]\[ V_{\text{half}} = \frac{1}{3} \times 900 \, \text{cm}^2 \times 15 \, \text{cm} = 4500 \, \text{cm}^3 \][/tex]
Step 7: Round the volume to the nearest cubic centimeter.
The volume when filled to half height is:
[tex]\[ 4500 \, \text{cm}^3 \][/tex]
So, the approximate volume of soil that Amira needs is:
[tex]\[ 4500 \, \text{cm}^3 \][/tex]
