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)

Answer 1

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

$$V_{\text{rectangular prism}}=30000,{\text{cm}}^3$$

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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

$$V_{\text{cylinder}}\approx 35325,{\text{cm}}^3$$

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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.