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 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)
inches tall and 8 inches in diameter. How many full pails of sand did you dig to carve out the pool?(1 point)
Answers
GPT-4o mini
To find out how many full pails of sand are needed 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 circular pool.
The volume \( V \) of a cylinder is calculated using the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height (or depth in this case).
1. The diameter of the pool is 3 feet, so the radius \( r \) is:
\[
r = \frac{3 \text{ feet}}{2} = 1.5 \text{ feet}
\]
2. The depth \( h \) of the pool is 18 inches, which we need to convert to feet:
\[
h = \frac{18 \text{ inches}}{12} = 1.5 \text{ feet}
\]
3. Now, we can calculate the volume of the pool:
\[
V = \pi (1.5 \text{ feet})^2 (1.5 \text{ feet}) = \pi (2.25 \text{ feet}^2) (1.5 \text{ feet}) = \pi (3.375 \text{ feet}^3)
\]
Approximating \(\pi\) as 3.14:
\[
V \approx 3.14 \times 3.375 \approx 10.59875 \text{ cubic feet}
\]
### Step 2: Calculate the volume of the pail.
The radius of the pail is:
\[
r = \frac{8 \text{ inches}}{2} = 4 \text{ inches}
\]
Convert this to feet:
\[
r = \frac{4}{12} = \frac{1}{3} \text{ feet}
\]
The height of the pail is:
\[
h = 13.5 \text{ inches} = \frac{13.5}{12} = 1.125 \text{ feet}
\]
Now, we can calculate the volume of the pail:
\[
V = \pi r^2 h = \pi \left( \frac{1}{3} \text{ feet} \right)^2 (1.125 \text{ feet}) = \pi \left( \frac{1}{9} \text{ feet}^2 \right) (1.125 \text{ feet}) = \pi \left( \frac{1.125}{9} \text{ feet}^3 \right)
\]
Calculating \(\frac{1.125}{9} = 0.125\):
\[
V \approx \pi (0.125 \text{ feet}^3) \approx 3.14 \times 0.125 \approx 0.3925 \text{ cubic feet}
\]
### Step 3: Calculate the number of full pails needed.
Now we can find the total number of pails needed:
\[
\text{Number of pails} = \frac{\text{Volume of pool}}{\text{Volume of pail}} = \frac{10.59875 \text{ cubic feet}}{0.3925 \text{ cubic feet}} \approx 26.98
\]
Since we are looking for full pails, we round this down. Therefore, the number of full pails of sand you dug to carve out the pool is:
\[
\boxed{26}
\]
### Step 1: Calculate the volume of the circular pool.
The volume \( V \) of a cylinder is calculated using the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height (or depth in this case).
1. The diameter of the pool is 3 feet, so the radius \( r \) is:
\[
r = \frac{3 \text{ feet}}{2} = 1.5 \text{ feet}
\]
2. The depth \( h \) of the pool is 18 inches, which we need to convert to feet:
\[
h = \frac{18 \text{ inches}}{12} = 1.5 \text{ feet}
\]
3. Now, we can calculate the volume of the pool:
\[
V = \pi (1.5 \text{ feet})^2 (1.5 \text{ feet}) = \pi (2.25 \text{ feet}^2) (1.5 \text{ feet}) = \pi (3.375 \text{ feet}^3)
\]
Approximating \(\pi\) as 3.14:
\[
V \approx 3.14 \times 3.375 \approx 10.59875 \text{ cubic feet}
\]
### Step 2: Calculate the volume of the pail.
The radius of the pail is:
\[
r = \frac{8 \text{ inches}}{2} = 4 \text{ inches}
\]
Convert this to feet:
\[
r = \frac{4}{12} = \frac{1}{3} \text{ feet}
\]
The height of the pail is:
\[
h = 13.5 \text{ inches} = \frac{13.5}{12} = 1.125 \text{ feet}
\]
Now, we can calculate the volume of the pail:
\[
V = \pi r^2 h = \pi \left( \frac{1}{3} \text{ feet} \right)^2 (1.125 \text{ feet}) = \pi \left( \frac{1}{9} \text{ feet}^2 \right) (1.125 \text{ feet}) = \pi \left( \frac{1.125}{9} \text{ feet}^3 \right)
\]
Calculating \(\frac{1.125}{9} = 0.125\):
\[
V \approx \pi (0.125 \text{ feet}^3) \approx 3.14 \times 0.125 \approx 0.3925 \text{ cubic feet}
\]
### Step 3: Calculate the number of full pails needed.
Now we can find the total number of pails needed:
\[
\text{Number of pails} = \frac{\text{Volume of pool}}{\text{Volume of pail}} = \frac{10.59875 \text{ cubic feet}}{0.3925 \text{ cubic feet}} \approx 26.98
\]
Since we are looking for full pails, we round this down. Therefore, the number of full pails of sand you dug to carve out the pool is:
\[
\boxed{26}
\]