Question
At the beach, you use your pail to dig a circular pool in the sand 3 feet in diameter and 18 inches deep. The pail is 13 and 1/2
inches tall and 8 inches in diameter. How many full pails of sand did you dig to carve out the pool?(1 point)
Responses
27 pails
27 pails
6 pails
6 pails
21 pails
21 pails
108 pails
inches tall and 8 inches in diameter. How many full pails of sand did you dig to carve out the pool?(1 point)
Responses
27 pails
27 pails
6 pails
6 pails
21 pails
21 pails
108 pails
Answers
GPT-4o mini
To determine how many full pails of sand you dug to carve out the pool, we first need to calculate the volume of the circular pool and the volume of the pail.
**Step 1: Calculate the volume of the pool.**
The pool is a cylinder with a diameter of 3 feet and a depth of 18 inches.
- Convert the diameter to feet:
- Diameter = 3 feet
- Radius = 3 feet / 2 = 1.5 feet
- Convert the depth to feet:
- Depth = 18 inches = 18/12 feet = 1.5 feet
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
Where:
- \( r = \) radius
- \( h = \) height (or depth)
Substituting the values:
\[
V = \pi (1.5)^2 (1.5) = \pi (2.25)(1.5) = \pi (3.375)
\]
Thus,
\[
V \approx 3.375 \times 3.14159 \approx 10.61 \text{ cubic feet}
\]
**Step 2: Calculate the volume of the pail.**
The pail has a diameter of 8 inches, so the radius is 4 inches.
- Convert the radius to feet:
- Radius = 4 inches = 4/12 feet = 1/3 feet
- The height of the pail is 13.5 inches = 13.5/12 feet = 1.125 feet
Using the volume formula for the cylinder again:
\[
V = \pi r^2 h
\]
Substituting the values:
\[
V = \pi \left(\frac{1}{3}\right)^2 (1.125) = \pi \left(\frac{1}{9}\right)(1.125) = \frac{1.125\pi}{9}
\]
Calculating the numerical value:
\[
V \approx \frac{1.125 \times 3.14159}{9} \approx \frac{3.53429}{9} \approx 0.3927 \text{ cubic feet}
\]
**Step 3: Calculate how many pails of sand were dug out for the pool.**
Now, we find the number of pails by dividing the volume of the pool by the volume of the pail:
\[
\text{Number of pails} = \frac{\text{Volume of pool}}{\text{Volume of pail}} = \frac{10.61}{0.3927} \approx 27
\]
Therefore, the answer is **27 pails.**
**Step 1: Calculate the volume of the pool.**
The pool is a cylinder with a diameter of 3 feet and a depth of 18 inches.
- Convert the diameter to feet:
- Diameter = 3 feet
- Radius = 3 feet / 2 = 1.5 feet
- Convert the depth to feet:
- Depth = 18 inches = 18/12 feet = 1.5 feet
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
Where:
- \( r = \) radius
- \( h = \) height (or depth)
Substituting the values:
\[
V = \pi (1.5)^2 (1.5) = \pi (2.25)(1.5) = \pi (3.375)
\]
Thus,
\[
V \approx 3.375 \times 3.14159 \approx 10.61 \text{ cubic feet}
\]
**Step 2: Calculate the volume of the pail.**
The pail has a diameter of 8 inches, so the radius is 4 inches.
- Convert the radius to feet:
- Radius = 4 inches = 4/12 feet = 1/3 feet
- The height of the pail is 13.5 inches = 13.5/12 feet = 1.125 feet
Using the volume formula for the cylinder again:
\[
V = \pi r^2 h
\]
Substituting the values:
\[
V = \pi \left(\frac{1}{3}\right)^2 (1.125) = \pi \left(\frac{1}{9}\right)(1.125) = \frac{1.125\pi}{9}
\]
Calculating the numerical value:
\[
V \approx \frac{1.125 \times 3.14159}{9} \approx \frac{3.53429}{9} \approx 0.3927 \text{ cubic feet}
\]
**Step 3: Calculate how many pails of sand were dug out for the pool.**
Now, we find the number of pails by dividing the volume of the pool by the volume of the pail:
\[
\text{Number of pails} = \frac{\text{Volume of pool}}{\text{Volume of pail}} = \frac{10.61}{0.3927} \approx 27
\]
Therefore, the answer is **27 pails.**