Question

a gardener uses a tray of 6 conical pots to plant seeds. each conical pot has a radius of 3 centimeters and a depth of 8 centimeters.

about how many cubic centimeters of soil are needed to plant the full tray? round to the nearest cubic centimeter.

(a) \\(226\text{ cm}^3\\)
(b) \\(301\text{ cm}^3\\)
(c) \\(452\text{ cm}^3\\)
(d) \\(678\text{ cm}^3\\)

Explanation:

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<reasoning>

Identify the given values and formulas

We need to find the total volume of soil required for a tray containing 6 identical conical pots.

• Number of pots, \(n = 6\)
• Radius of each cone, \(r = 3\text{ cm}\)
• Depth (height) of each cone, \(h = 8\text{ cm}\)

The formula for the volume of a single cone is:
\[V_{\text{cone}} = \frac{1}{3}\pi r^2 h\]

Calculate the volume of one conical pot

Substitute the given values \(r = 3\) and \(h = 8\) into the cone volume formula:
\[V_{\text{cone}} = \frac{1}{3}\pi (3)^2 (8)\]
\[V_{\text{cone}} = \frac{1}{3}\pi (9)(8)\]
\[V_{\text{cone}} = 24\pi\text{ cm}^3\]

Calculate the total volume for the full tray

Since there are 6 identical pots in the tray, we multiply the volume of one pot by 6:
\[V_{\text{total}} = 6 \times V_{\text{cone}}\]
\[V_{\text{total}} = 6 \times 24\pi\]
\[V_{\text{total}} = 144\pi\text{ cm}^3\]

Round to the nearest cubic centimeter

Using the approximation \(\pi \approx 3.14159\):
\[V_{\text{total}} \approx 144 \times 3.14159\]
\[V_{\text{total}} \approx 452.39\text{ cm}^3\]

Rounding to the nearest cubic centimeter gives \(452\text{ cm}^3\).
</reasoning>

<answer>
<mcq-option>226 cm³</mcq-option>
<mcq-option>301 cm³</mcq-option>
<mcq-correct>452 cm³</mcq-correct>
<mcq-option>678 cm³</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given values and formulas

We need to find the total volume of soil required for a tray containing 6 identical conical pots.

• Number of pots, \(n = 6\)
• Radius of each cone, \(r = 3\text{ cm}\)
• Depth (height) of each cone, \(h = 8\text{ cm}\)

The formula for the volume of a single cone is:
\[V_{\text{cone}} = \frac{1}{3}\pi r^2 h\]

Calculate the volume of one conical pot

Substitute the given values \(r = 3\) and \(h = 8\) into the cone volume formula:
\[V_{\text{cone}} = \frac{1}{3}\pi (3)^2 (8)\]
\[V_{\text{cone}} = \frac{1}{3}\pi (9)(8)\]
\[V_{\text{cone}} = 24\pi\text{ cm}^3\]

Calculate the total volume for the full tray

Since there are 6 identical pots in the tray, we multiply the volume of one pot by 6:
\[V_{\text{total}} = 6 \times V_{\text{cone}}\]
\[V_{\text{total}} = 6 \times 24\pi\]
\[V_{\text{total}} = 144\pi\text{ cm}^3\]

Round to the nearest cubic centimeter

Using the approximation \(\pi \approx 3.14159\):
\[V_{\text{total}} \approx 144 \times 3.14159\]
\[V_{\text{total}} \approx 452.39\text{ cm}^3\]

Rounding to the nearest cubic centimeter gives \(452\text{ cm}^3\).
</reasoning>

<answer>
<mcq-option>226 cm³</mcq-option>
<mcq-option>301 cm³</mcq-option>
<mcq-correct>452 cm³</mcq-correct>
<mcq-option>678 cm³</mcq-option>
</answer>

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