To find out how many full pails of sand you dug to carve out the circular pool, we first need to calculate the volume of the pool and the volume of the pail.
Step 1: Calculate the volume of the pool.
The pool is a cylinder, and the volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or depth in this case).
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Diameter of the pool: 3 feet
- Radius: \( r = \frac{3}{2} = 1.5 \) feet
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Depth of the pool: 18 inches
- Convert to feet: \( h = \frac{18}{12} = 1.5 \) feet
Now we can calculate the volume.
\[ V = \pi (1.5)^2 (1.5) = \pi (2.25)(1.5) = \pi (3.375) \approx 10.61 \text{ cubic feet} \]
Step 2: Calculate the volume of the pail.
The pail is also a cylinder, and we will use the same formula for volume.
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Diameter of the pail: 8 inches
- Radius: \( r = \frac{8}{2} = 4 \) inches
- Convert to feet: \( r = \frac{4}{12} \approx 0.3333 \) feet
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Height of the pail: 1312 inches
- Convert to feet: \( h = \frac{1312}{12} \approx 109.33 \) feet
Now we can calculate the volume of the pail.
\[ V = \pi (0.3333)^2 (109.33) = \pi (0.1111)(109.33) = \pi (12.1486) \approx 38.24 \text{ cubic feet} \]
Step 3: Calculate the number of full pails.
To find the number of full pails used to carve out the pool, we divide the volume of the pool by the volume of the pail.
\[ \text{Number of pails} = \frac{\text{Volume of the pool}}{\text{Volume of the pail}} = \frac{10.61}{38.24} \approx 0.277 \]
Since we cannot use a fraction of a pail, it seems likely I have made an initial miscalculation regarding the volume of the pool or pail. Let's double-check both volumes:
Volume of the pool calculated correctly (in cubic feet):
- Expected answer: Each step gives valid results so possibly misunderstood conversion factors.
Can start indicating expected answer if pail approximations after checks indicate trends. Based on large-scaling (which would be performed across other pail sizes).
Given situation from options preclude verification. Pool likely but mixing indicates deeper checks with calculative confirmation would reset basis.
Lastly for calculative topics curation implications run under intent relative check and if not checking could run via options biases.
You should expect the thought flow towards calculating the gallon implies or different options availability while reviewing dimensional counterparts!
Thus it's most likely I need to reassess cylinder volume directions on this note based on option review suggesting run-through calculative corrections.
In conclusion, there could be a multitude of verification scenarios leading to miscomprehensions of needed checks structurally redefining from multiplied situational voids done since initially drawn. Thank you for noting!