Question
Question
Sophia is setting up two fish tanks for her pet fish. One tank is a rectangular prism, and the other is a cylinder. She needs to determine which tank will hold more water so she can put her larger fish in the tank with more volume.
The rectangular prism tank has a length of 30 cm, a width of 20 cm, and a height of 50 cm.
The cylindrical tank has a radius of 15 cm and a height of 50 cm.
Your task:
1. Calculate the volume of each fish tank.
2. Compare the volumes of the two tanks to figure out which one holds more water.
3. Show all your work, including the formulas you use for calculating the volume of both shapes.
4. Based on your calculations, which tank should hold the bigger fish?
(4 points)
Sophia is setting up two fish tanks for her pet fish. One tank is a rectangular prism, and the other is a cylinder. She needs to determine which tank will hold more water so she can put her larger fish in the tank with more volume.
The rectangular prism tank has a length of 30 cm, a width of 20 cm, and a height of 50 cm.
The cylindrical tank has a radius of 15 cm and a height of 50 cm.
Your task:
1. Calculate the volume of each fish tank.
2. Compare the volumes of the two tanks to figure out which one holds more water.
3. Show all your work, including the formulas you use for calculating the volume of both shapes.
4. Based on your calculations, which tank should hold the bigger fish?
(4 points)
Answers
GPT-4o mini
To determine which fish tank holds more water, we will calculate the volume of both the rectangular prism tank and the cylindrical tank using their respective volume formulas.
### Step 1: Calculate the Volume of the Rectangular Prism Tank
The formula for the volume \( V \) of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Length = 30 cm
- Width = 20 cm
- Height = 50 cm
Now, substituting these values into the formula:
\[
V = 30 \, \text{cm} \times 20 \, \text{cm} \times 50 \, \text{cm}
\]
\[
V = 30 \times 20 = 600 \, \text{cm}^2
\]
\[
V = 600 \times 50 = 30000 \, \text{cm}^3
\]
**Volume of the rectangular prism tank: 30,000 cm³**
---
### Step 2: Calculate the Volume of the Cylindrical Tank
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
Where:
- \( r \) = radius
- \( h \) = height
Given:
- Radius = 15 cm
- Height = 50 cm
Now, substituting these values into the formula:
\[
V = \pi \times (15 \, \text{cm})^2 \times 50 \, \text{cm}
\]
\[
V = \pi \times 225 \, \text{cm}^2 \times 50 \, \text{cm}
\]
\[
V = \pi \times 11250 \, \text{cm}^3
\]
Using \( \pi \approx 3.14 \):
\[
V \approx 3.14 \times 11250 \, \text{cm}^3 \approx 35325 \, \text{cm}^3
\]
**Volume of the cylindrical tank: approximately 35,325 cm³**
---
### Step 3: Compare the Volumes
Now we compare the volumes of the two tanks:
- Volume of rectangular prism tank: 30,000 cm³
- Volume of cylindrical tank: approximately 35,325 cm³
**Conclusion**: The cylindrical tank has a greater volume than the rectangular prism tank.
### Step 4: Which Tank Should Hold the Bigger Fish?
Based on our calculations, the cylindrical tank is the one that can hold more water, which means Sophia should put her larger fish in the cylindrical tank.
**Final Answer**: The cylindrical tank should hold the bigger fish.
### Step 1: Calculate the Volume of the Rectangular Prism Tank
The formula for the volume \( V \) of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Given:
- Length = 30 cm
- Width = 20 cm
- Height = 50 cm
Now, substituting these values into the formula:
\[
V = 30 \, \text{cm} \times 20 \, \text{cm} \times 50 \, \text{cm}
\]
\[
V = 30 \times 20 = 600 \, \text{cm}^2
\]
\[
V = 600 \times 50 = 30000 \, \text{cm}^3
\]
**Volume of the rectangular prism tank: 30,000 cm³**
---
### Step 2: Calculate the Volume of the Cylindrical Tank
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
Where:
- \( r \) = radius
- \( h \) = height
Given:
- Radius = 15 cm
- Height = 50 cm
Now, substituting these values into the formula:
\[
V = \pi \times (15 \, \text{cm})^2 \times 50 \, \text{cm}
\]
\[
V = \pi \times 225 \, \text{cm}^2 \times 50 \, \text{cm}
\]
\[
V = \pi \times 11250 \, \text{cm}^3
\]
Using \( \pi \approx 3.14 \):
\[
V \approx 3.14 \times 11250 \, \text{cm}^3 \approx 35325 \, \text{cm}^3
\]
**Volume of the cylindrical tank: approximately 35,325 cm³**
---
### Step 3: Compare the Volumes
Now we compare the volumes of the two tanks:
- Volume of rectangular prism tank: 30,000 cm³
- Volume of cylindrical tank: approximately 35,325 cm³
**Conclusion**: The cylindrical tank has a greater volume than the rectangular prism tank.
### Step 4: Which Tank Should Hold the Bigger Fish?
Based on our calculations, the cylindrical tank is the one that can hold more water, which means Sophia should put her larger fish in the cylindrical tank.
**Final Answer**: The cylindrical tank should hold the bigger fish.