Question
Boris puts water for his cat in a small, round bowl. He puts water for his dog in a large, round bowl. After the animals drink, each bowl is 1 by 8 full of water. Do they contain the same amount? Explain.
Answers
GPT-4o mini
To determine whether the small round bowl for the cat and the large round bowl for the dog contain the same amount of water after drinking, we need to look at their shapes and how the volume of water relates to their dimensions.
The cat's bowl is small, while the dog's bowl is large. However, both bowls are round. Assuming that they are both cylindrical in shape, the volume of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
Now, after the animals drink, both bowls are \( \frac{1}{8} \) full of water. This means that regardless of the sizes of the bowls, each bowl has the same fraction of its total volume filled with water.
However, the actual volume of water in each bowl depends on the original volumes of the bowls. If we denote the volume of the small bowl as \( V_{cat} \) and the volume of the large bowl as \( V_{dog} \), the remaining water can be calculated as follows:
\[
\text{Water in cat's bowl} = \frac{1}{8} V_{cat}
\]
\[
\text{Water in dog's bowl} = \frac{1}{8} V_{dog}
\]
Since \( V_{cat} \) is smaller than \( V_{dog} \) (because the cat's bowl is small and the dog's bowl is large), it follows that:
\[
\frac{1}{8} V_{cat} < \frac{1}{8} V_{dog}
\]
Therefore, the amount of water remaining in the cat's bowl is less than the amount of water remaining in the dog's bowl. In conclusion, even though both bowls are \( \frac{1}{8} \) full, they do not contain the same amount of water. The dog's bowl contains more water than the cat's bowl.
The cat's bowl is small, while the dog's bowl is large. However, both bowls are round. Assuming that they are both cylindrical in shape, the volume of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
Now, after the animals drink, both bowls are \( \frac{1}{8} \) full of water. This means that regardless of the sizes of the bowls, each bowl has the same fraction of its total volume filled with water.
However, the actual volume of water in each bowl depends on the original volumes of the bowls. If we denote the volume of the small bowl as \( V_{cat} \) and the volume of the large bowl as \( V_{dog} \), the remaining water can be calculated as follows:
\[
\text{Water in cat's bowl} = \frac{1}{8} V_{cat}
\]
\[
\text{Water in dog's bowl} = \frac{1}{8} V_{dog}
\]
Since \( V_{cat} \) is smaller than \( V_{dog} \) (because the cat's bowl is small and the dog's bowl is large), it follows that:
\[
\frac{1}{8} V_{cat} < \frac{1}{8} V_{dog}
\]
Therefore, the amount of water remaining in the cat's bowl is less than the amount of water remaining in the dog's bowl. In conclusion, even though both bowls are \( \frac{1}{8} \) full, they do not contain the same amount of water. The dog's bowl contains more water than the cat's bowl.