Answer :
To find the number of buckets of soil needed to fill the rectangular planter, you need to first find the volumes of both the bucket and the planter, and then determine how many such buckets are required to fill the planter.
### Step-by-Step Solution:
#### 1. Convert Dimensions to a Consistent Unit
- The dimensions of the planter are given in meters and centimeters, so first, we convert the length from meters to centimeters.
- Length of the planter: 1.2 meters = 1.2 * 100 = 120 cm
#### 2. Calculate the Volume of the Bucket (Cylinder)
To find the volume of the bucket, we use the formula for the volume of a cylinder:
[tex]\[ V_{\text{cylinder}} = \pi \cdot r^2 \cdot h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cylinder (bucket)
- [tex]\( h \)[/tex] is the height of the cylinder (bucket)
- Given: [tex]\( r = 18 \)[/tex] cm, [tex]\( h = 18 \)[/tex] cm
Using these values:
[tex]\[ V_{\text{bucket}} = \pi \cdot (18^2) \cdot 18 \approx 18321.77 \, \text{cm}^3 \][/tex]
#### 3. Calculate the Volume of the Planter (Rectangular Prism)
To find the volume of the planter, use the formula for the volume of a rectangular prism:
[tex]\[ V_{\text{prism}} = \text{length} \cdot \text{width} \cdot \text{height} \][/tex]
Given dimensions:
- Length = 120 cm
- Width = 40 cm
- Depth = 20 cm
Using these values:
[tex]\[ V_{\text{planter}} = 120 \cdot 40 \cdot 20 = 96000 \, \text{cm}^3 \][/tex]
#### 4. Calculate the Number of Buckets Needed
To find out how many buckets are needed to fill the planter, divide the volume of the planter by the volume of one bucket:
[tex]\[ \text{Number of buckets} = \frac{V_{\text{planter}}}{V_{\text{bucket}}} = \frac{96000}{18321.77} \approx 5.24 \][/tex]
### Conclusion
The landscaper will need approximately 5.24 buckets of soil to fill the rectangular planter. Since you cannot practically have a fraction of a bucket, you would need 6 buckets to ensure you have enough soil to completely fill the planter.
### Step-by-Step Solution:
#### 1. Convert Dimensions to a Consistent Unit
- The dimensions of the planter are given in meters and centimeters, so first, we convert the length from meters to centimeters.
- Length of the planter: 1.2 meters = 1.2 * 100 = 120 cm
#### 2. Calculate the Volume of the Bucket (Cylinder)
To find the volume of the bucket, we use the formula for the volume of a cylinder:
[tex]\[ V_{\text{cylinder}} = \pi \cdot r^2 \cdot h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cylinder (bucket)
- [tex]\( h \)[/tex] is the height of the cylinder (bucket)
- Given: [tex]\( r = 18 \)[/tex] cm, [tex]\( h = 18 \)[/tex] cm
Using these values:
[tex]\[ V_{\text{bucket}} = \pi \cdot (18^2) \cdot 18 \approx 18321.77 \, \text{cm}^3 \][/tex]
#### 3. Calculate the Volume of the Planter (Rectangular Prism)
To find the volume of the planter, use the formula for the volume of a rectangular prism:
[tex]\[ V_{\text{prism}} = \text{length} \cdot \text{width} \cdot \text{height} \][/tex]
Given dimensions:
- Length = 120 cm
- Width = 40 cm
- Depth = 20 cm
Using these values:
[tex]\[ V_{\text{planter}} = 120 \cdot 40 \cdot 20 = 96000 \, \text{cm}^3 \][/tex]
#### 4. Calculate the Number of Buckets Needed
To find out how many buckets are needed to fill the planter, divide the volume of the planter by the volume of one bucket:
[tex]\[ \text{Number of buckets} = \frac{V_{\text{planter}}}{V_{\text{bucket}}} = \frac{96000}{18321.77} \approx 5.24 \][/tex]
### Conclusion
The landscaper will need approximately 5.24 buckets of soil to fill the rectangular planter. Since you cannot practically have a fraction of a bucket, you would need 6 buckets to ensure you have enough soil to completely fill the planter.
