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:

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 \(452\text{ cm}^3\).

Answer:

• 226 cm³
• 301 cm³
• 452 cm³ (Correct answer)
• 678 cm³