Question
A circular, inflatable kiddie pool needs to be filled with water, but the hose does not reach. You have a bucket 1 foot in diameter and 15 inches tall. Determine the number of whole buckets needed to fill the swimming pool if the pool’s diameter is 12 feet and it is 30 inches tall. Each bucket is filled to 3 inches from the top to prevent spilling. Additionally, you want to stop filling the pool when the water level is 1 foot from the top so that the water doesn’t spill over the edges.(1 point)
Answers
GPT-4o mini
To determine the number of whole buckets needed to fill the kiddie pool, we will first calculate the volume of water the kiddie pool can hold up to the desired level and then find the volume of water that one bucket can hold.
### Step 1: Volume of the Kiddie Pool
The diameter of the kiddie pool is 12 feet, so the radius \( r \) of the pool is:
\[
r = \frac{12 \text{ feet}}{2} = 6 \text{ feet}
\]
The pool height is 30 inches, which we need to convert into feet:
\[
30 \text{ inches} = \frac{30}{12} \text{ feet} = 2.5 \text{ feet}
\]
We need to stop filling the pool when the water level is 1 foot from the top. Therefore, the height of the water will be:
\[
\text{Height of water} = 2.5 \text{ feet} - 1 \text{ foot} = 1.5 \text{ feet}
\]
Now, we can calculate the volume of water in the pool using the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
Substituting in our values, we have:
\[
V = \pi (6 \text{ feet})^2 (1.5 \text{ feet}) = \pi (36 \text{ feet}^2)(1.5 \text{ feet}) = 54\pi \text{ cubic feet}
\]
### Step 2: Volume of the Bucket
The bucket has a diameter of 1 foot, giving it a radius \( r \):
\[
r = \frac{1 \text{ foot}}{2} = 0.5 \text{ feet}
\]
The height of the bucket is 15 inches, which we convert to feet:
\[
15 \text{ inches} = \frac{15}{12} \text{ feet} = 1.25 \text{ feet}
\]
We will fill the bucket to 3 inches from the top. Therefore, the height of the water in the bucket is:
\[
\text{Height of water} = 1.25 \text{ feet} - \frac{3}{12} \text{ feet} = 1.25 \text{ feet} - 0.25 \text{ feet} = 1 \text{ foot}
\]
Now, we can calculate the volume of water in the bucket:
\[
V = \pi r^2 h
\]
Substituting in our values, we have:
\[
V = \pi (0.5 \text{ feet})^2 (1 \text{ foot}) = \pi (0.25 \text{ feet}^2)(1 \text{ foot}) = 0.25\pi \text{ cubic feet}
\]
### Step 3: Calculate the Number of Buckets Needed
Now we need to find how many buckets are needed to fill the kiddie pool:
\[
\text{Number of buckets} = \frac{\text{Volume of kiddie pool}}{\text{Volume of one bucket}} = \frac{54\pi \text{ cubic feet}}{0.25\pi \text{ cubic feet}} = \frac{54}{0.25} = 216
\]
Therefore, the number of whole buckets needed to fill the swimming pool is:
\[
\boxed{216}
\]
### Step 1: Volume of the Kiddie Pool
The diameter of the kiddie pool is 12 feet, so the radius \( r \) of the pool is:
\[
r = \frac{12 \text{ feet}}{2} = 6 \text{ feet}
\]
The pool height is 30 inches, which we need to convert into feet:
\[
30 \text{ inches} = \frac{30}{12} \text{ feet} = 2.5 \text{ feet}
\]
We need to stop filling the pool when the water level is 1 foot from the top. Therefore, the height of the water will be:
\[
\text{Height of water} = 2.5 \text{ feet} - 1 \text{ foot} = 1.5 \text{ feet}
\]
Now, we can calculate the volume of water in the pool using the formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
Substituting in our values, we have:
\[
V = \pi (6 \text{ feet})^2 (1.5 \text{ feet}) = \pi (36 \text{ feet}^2)(1.5 \text{ feet}) = 54\pi \text{ cubic feet}
\]
### Step 2: Volume of the Bucket
The bucket has a diameter of 1 foot, giving it a radius \( r \):
\[
r = \frac{1 \text{ foot}}{2} = 0.5 \text{ feet}
\]
The height of the bucket is 15 inches, which we convert to feet:
\[
15 \text{ inches} = \frac{15}{12} \text{ feet} = 1.25 \text{ feet}
\]
We will fill the bucket to 3 inches from the top. Therefore, the height of the water in the bucket is:
\[
\text{Height of water} = 1.25 \text{ feet} - \frac{3}{12} \text{ feet} = 1.25 \text{ feet} - 0.25 \text{ feet} = 1 \text{ foot}
\]
Now, we can calculate the volume of water in the bucket:
\[
V = \pi r^2 h
\]
Substituting in our values, we have:
\[
V = \pi (0.5 \text{ feet})^2 (1 \text{ foot}) = \pi (0.25 \text{ feet}^2)(1 \text{ foot}) = 0.25\pi \text{ cubic feet}
\]
### Step 3: Calculate the Number of Buckets Needed
Now we need to find how many buckets are needed to fill the kiddie pool:
\[
\text{Number of buckets} = \frac{\text{Volume of kiddie pool}}{\text{Volume of one bucket}} = \frac{54\pi \text{ cubic feet}}{0.25\pi \text{ cubic feet}} = \frac{54}{0.25} = 216
\]
Therefore, the number of whole buckets needed to fill the swimming pool is:
\[
\boxed{216}
\]