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Anderson’s Nursery was choosing between two different greenhouses.

2 greenhouses have a rectangular prism base with a length of 12 meters, width of 12 meters, and height of 10 meters. Greenhouse 1 has a square pyramid top with a base of 12 meters by 12 meters and height of 8 meters. Greenhouse 2 has a triangular prism top. The triangular sides have a base of 12 meters and height of 8 meters. The prism has a height of 12 meters.

Which statements are true about the greenhouses? Select two options.
The space inside greenhouse 1 can be found using the equation V = 12 (12) (10) + one-half (12) (12) (8).
The space inside greenhouse 1 can be found using the equation V = 12 (12) (10) + one-third (12) (12) (8).
The space inside greenhouse 2 can be found using the equation V = 12 (12) (10) + one-half (12) (12) (8).
The space inside greenhouse 2 can be found using the equation V = 12 (12) (10) + one-third (12) (12) (8).
The space inside greenhouse 1 is 192 meters cubed greater than the space inside greenhouse 2.

1 answer

To find the volume of the two greenhouses, we will look at the structures individually.

Greenhouse 1 (Square Pyramid on top of Rectangular Prism):

  1. The volume of the rectangular prism base is calculated using the formula \( V = \text{length} \times \text{width} \times \text{height} \). \[ V_{prism} = 12 \times 12 \times 10 = 1440 , \text{m}^3 \]

  2. The volume of the square pyramid on top is calculated using the formula \( V = \frac{1}{3} \times \text{base area} \times \text{height} \).

    • The base area of the pyramid is \( 12 \times 12 = 144 , \text{m}^2 \). \[ V_{pyramid} = \frac{1}{3} \times 144 \times 8 = 384 , \text{m}^3 \]

  3. The total volume of Greenhouse 1 is: \[ V_{1} = V_{prism} + V_{pyramid} = 1440 + 384 = 1824 , \text{m}^3 \]

Greenhouse 2 (Triangular Prism on top of Rectangular Prism):

  1. The volume of the rectangular prism base remains the same: \[ V_{prism} = 12 \times 12 \times 10 = 1440 , \text{m}^3 \]

  2. The volume of the triangular prism on top is calculated using the formula \( V = \text{base area} \times \text{height} \).

    • The area of the triangular base is \(\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 8 = 48 , \text{m}^2\). \[ V_{triangular , prism} = \text{base area} \times \text{height} = 48 \times 12 = 576 , \text{m}^3 \]

  3. The total volume of Greenhouse 2 is: \[ V_{2} = V_{prism} + V_{triangular , prism} = 1440 + 576 = 2016 , \text{m}^3 \]

Now, let's evaluate the statements:

  1. The space inside greenhouse 1 can be found using the equation \( V = 12 (12) (10) + \frac{1}{2} (12) (12) (8) \).

    • This statement is False because the volume of the square pyramid must be calculated with \( \frac{1}{3} \), not \( \frac{1}{2} \).

  2. The space inside greenhouse 1 can be found using the equation \( V = 12 (12) (10) + \frac{1}{3} (12) (12) (8) \).

    • This statement is True since it correctly describes the volume calculation.

  3. The space inside greenhouse 2 can be found using the equation \( V = 12 (12) (10) + \frac{1}{2} (12) (12) (8) \).

    • This statement is False because the volume of the top does not involve using \( \frac{1}{2} \).

  4. The space inside greenhouse 2 can be found using the equation \( V = 12 (12) (10) + \frac{1}{3} (12) (12) (8) \).

    • This statement is False because we do not use \( \frac{1}{3} \) for the volume of the triangular prism.

  5. The space inside greenhouse 1 is 192 meters cubed greater than the space inside greenhouse 2.

    • This statement is False. The volumes are: \( V_{1} = 1824 , \text{m}^3 \) and \( V_{2} = 2016 , \text{m}^3 \). Greenhouse 2 is actually larger, not smaller.

Final answers:

  • The true statements are:
    • The space inside greenhouse 1 can be found using the equation \( V = 12 (12) (10) + \frac{1}{3} (12) (12) (8) \).