- How many liters are contained in a 5 gallon bucket?
- What is the volume of the 5 gallon bucket in cm³?
- If a 5 gallon bucket could be modeled as a perfect cylinder with a diameter of 30 cm, how tall (in cm is the bucket?
- If you pour one (1) gallon of water into the 5 gallon bucket described in Question #2, how high (in cm) will the water reach in the bucket?
- If the bucket now holds 5 gallons, how much would it weigh in lbs? Assume an empty bucket has a mass of 1 kg.
The Correct Answer and Explanation is :
Let’s go through each question step by step:
1. How many liters are contained in a 5-gallon bucket?
- 1 gallon ≈ 3.785 liters.
- Therefore, a 5-gallon bucket would contain:
[
5 \, \text{gallons} \times 3.785 \, \frac{\text{liters}}{\text{gallon}} = 18.925 \, \text{liters}.
]
Answer: 18.925 liters.
2. What is the volume of the 5-gallon bucket in cm³?
Since the volume of 5 gallons is 18.925 liters, and 1 liter = 1000 cm³, the volume in cubic centimeters (cm³) is:
[
18.925 \, \text{liters} \times 1000 \, \frac{\text{cm}^3}{\text{liter}} = 18,925 \, \text{cm}^3.
]
Answer: 18,925 cm³.
3. If a 5-gallon bucket could be modeled as a perfect cylinder with a diameter of 30 cm, how tall (in cm) is the bucket?
We can use the formula for the volume of a cylinder:
[
V = \pi r^2 h
]
Where:
- ( V = 18,925 \, \text{cm}^3 ) (from the previous calculation),
- ( r = \frac{\text{diameter}}{2} = \frac{30}{2} = 15 \, \text{cm} ).
We rearrange the formula to solve for ( h ) (the height of the bucket):
[
h = \frac{V}{\pi r^2} = \frac{18,925}{\pi \times 15^2} = \frac{18,925}{\pi \times 225} \approx \frac{18,925}{706.86} \approx 26.8 \, \text{cm}.
]
Answer: The height of the bucket is approximately 26.8 cm.
4. If you pour one (1) gallon of water into the 5-gallon bucket described in Question #2, how high (in cm) will the water reach in the bucket?
- The volume of 1 gallon = 3.785 liters = 3,785 cm³.
- We will use the same cylinder volume formula, but now the volume is 3,785 cm³, and we need to find the height ( h_{\text{water}} ).
[
h_{\text{water}} = \frac{V_{\text{water}}}{\pi r^2} = \frac{3,785}{\pi \times 15^2} = \frac{3,785}{\pi \times 225} \approx \frac{3,785}{706.86} \approx 5.4 \, \text{cm}.
]
Answer: The water will reach approximately 5.4 cm high in the bucket.
5. If the bucket now holds 5 gallons, how much would it weigh in lbs? Assume an empty bucket has a mass of 1 kg.
- The mass of 5 gallons of water is:
[
\text{Mass of water} = 5 \, \text{gallons} \times 8.34 \, \frac{\text{lbs}}{\text{gallon}} = 41.7 \, \text{lbs}.
] - The empty bucket weighs 1 kg, which is approximately 2.2 lbs.
- So, the total weight is:
[
\text{Total weight} = 41.7 \, \text{lbs} + 2.2 \, \text{lbs} = 43.9 \, \text{lbs}.
]
Answer: The total weight of the bucket when filled with 5 gallons of water is approximately 43.9 lbs.
300-Word Explanation:
To solve this problem, we applied basic volume and mass conversions. A 5-gallon bucket holds a volume of 18.925 liters, which is equivalent to 18,925 cubic centimeters (cm³), given that 1 liter = 1,000 cm³. To calculate the height of the bucket when modeled as a perfect cylinder, we used the formula for the volume of a cylinder: ( V = \pi r^2 h ), where ( V ) is the volume, ( r ) is the radius, and ( h ) is the height. By rearranging the formula to solve for ( h ), we found that the height of the 5-gallon bucket is about 26.8 cm.
When 1 gallon (or 3,785 cm³) of water is poured into the bucket, we calculated how high the water would reach using the same formula. This resulted in a height of about 5.4 cm. Next, we found the total weight of the bucket when filled with 5 gallons of water. The mass of 5 gallons of water is 41.7 lbs, and the mass of the empty bucket is 2.2 lbs (since 1 kg ≈ 2.2 lbs). Adding these together, the total weight is approximately 43.9 lbs.
This problem demonstrates how volume, mass, and dimensions are interconnected. It also shows the utility of applying simple formulas in real-world scenarios, such as determining the height of water in a cylindrical container or calculating weight. This approach is commonly used in physics, engineering, and everyday tasks like filling containers or designing objects.
Let me generate a simple diagram for you based on the bucket dimensions.
Here is the diagram of the cylindrical bucket with the given dimensions and water level. It shows the diameter, height of the bucket, and the height of the water when 1 gallon is poured in. Let me know if you need further clarification or adjustments!
