To determine the volumes of both fish tanks and compare them, we will use the formulas for the volumes of a rectangular prism and a cylinder.
Step 1: Calculate the Volume of the Rectangular Prism Tank
The formula for the volume \( V \) of a rectangular prism is given by: \[ V = \text{length} \times \text{width} \times \text{height} \]
For Sophia's rectangular prism tank:
- Length = 30 cm
- Width = 20 cm
- Height = 50 cm
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
\[ V_{\text{rectangular prism}} = 30000 , \text{cm}^3 \]
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:
- \( r \) is the radius
- \( h \) is the height
For Sophia's cylindrical tank:
- Radius = 15 cm
- Height = 50 cm
Substituting these values into the formula: \[ V = \pi (15 , \text{cm})^2 (50 , \text{cm}) \] Calculating \( (15 , \text{cm})^2 \): \[ 15^2 = 225 , \text{cm}^2 \]
Now substitute that value back into the volume formula: \[ V = \pi \times 225 , \text{cm}^2 \times 50 , \text{cm} \] \[ V = \pi \times 11250 , \text{cm}^3 \]
Using \( \pi \approx 3.14 \) for the calculation: \[ V \approx 3.14 \times 11250 , \text{cm}^3 \] \[ V \approx 35325 , \text{cm}^3 \]
Volume of the Cylindrical Tank
\[ V_{\text{cylinder}} \approx 35325 , \text{cm}^3 \]
Step 3: Compare the Volumes of the Two Tanks
Now that we have calculated both volumes, we can compare them:
- Volume of the rectangular prism tank: \( 30000 , \text{cm}^3 \)
- Volume of the cylindrical tank: \( 35325 , \text{cm}^3 \)
To determine which tank holds more water: \[ 35325 , \text{cm}^3 > 30000 , \text{cm}^3 \]
Conclusion
The cylindrical tank holds more water than the rectangular prism tank.
Step 4: Determine Which Tank Should Hold the Bigger Fish
Since the cylindrical tank has a larger volume, that would be the better choice for placing the larger fish.
Therefore, Sophia should put her larger fish in the cylindrical tank.