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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. [tex]\( 226 \, \text{cm}^3 \)[/tex]
B. [tex]\( 301 \, \text{cm}^3 \)[/tex]
C. [tex]\( 452 \, \text{cm}^3 \)[/tex]
D. [tex]\( 678 \, \text{cm}^3 \)[/tex]



Answer :

GinnyAnswer
To determine how many cubic centimeters of soil are needed to plant the full tray of conical pots, follow these steps:

1. Calculate the volume of a single conical pot:

The volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cone
- [tex]\( h \)[/tex] is the height (or depth) of the cone
- [tex]\( \pi \)[/tex] is approximately 3.14159

2. Substitute the given values into the formula:

- Radius [tex]\( r = 3 \)[/tex] cm
- Depth [tex]\( h = 8 \)[/tex] cm

Plug these values into the formula:
[tex]\[ V = \frac{1}{3} \pi (3)^2 (8) \][/tex]

3. Simplify and calculate the volume for one pot:

[tex]\[ V = \frac{1}{3} \pi (9) (8) = \frac{1}{3} \pi (72) \][/tex]
[tex]\[ V = 24 \pi \approx 24 \times 3.14159 = 75.39822368615503 \text{ cubic centimeters} \][/tex]

4. Calculate the total volume for 6 conical pots:

Since each pot has a volume of approximately 75.398 cubic centimeters and there are 6 pots:
[tex]\[ \text{Total Volume} = 75.39822368615503 \times 6 = 452.3893421169302 \text{ cubic centimeters} \][/tex]

5. Round the total volume to the nearest cubic centimeter:

[tex]\[ \text{Rounded Total Volume} \approx 452 \text{ cubic centimeters} \][/tex]

Hence, the total amount of soil needed to plant the full tray of conical pots is approximately [tex]\( 452 \)[/tex] cubic centimeters.

The correct answer is:
[tex]\[ \boxed{452 \text{ cm}^3} \][/tex]

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