Question
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.
Make sure to select 2 statements
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.
Make sure to select 2 statements
Answers
GPT-4o mini
Let's analyze the statements one by one for both greenhouse designs.
### Greenhouse 1:
- The base: rectangular prism with dimensions 12m x 12m x 10m
- The top: square pyramid with a base of 12m x 12m and height of 8m
The volume \( V \) of the rectangular prism can be calculated as:
\[
V_{rectangular \, prism} = \text{length} \times \text{width} \times \text{height} = 12 \times 12 \times 10 = 1440 \, \text{m}^3
\]
The volume \( V \) of the square pyramid can be calculated as:
\[
V_{pyramid} = \frac{1}{3} \times \text{base area} \times \text{height} = \frac{1}{3} \times (12 \times 12) \times 8 = \frac{1}{3} \times 144 \times 8 = \frac{1152}{3} = 384 \, \text{m}^3
\]
The total volume of Greenhouse 1:
\[
V_{greenhouse 1} = V_{rectangular \, prism} + V_{pyramid} = 1440 + 384 = 1824 \, \text{m}^3
\]
### Greenhouse 2:
- The base: rectangular prism with dimensions 12m x 12m x 10m
- The top: triangular prism with a triangular base of height 8m and base of 12m, and prism height of 12m
The volume \( V \) of the triangular base can be calculated as:
\[
\text{Area of the triangular base} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 8 = 48 \, \text{m}^2
\]
The volume \( V \) of the triangular prism is:
\[
V_{triangular \, prism} = \text{base area} \times \text{height} = 48 \times 12 = 576 \, \text{m}^3
\]
The total volume of Greenhouse 2:
\[
V_{greenhouse 2} = V_{rectangular \, prism} + V_{triangular \, prism} = 1440 + 576 = 2016 \, \text{m}^3
\]
### Evaluating the Statements:
Now let's analyze the provided statements:
1. **The space inside greenhouse 1 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{2} \times 12 \times 12 \times 8 \).**
**(False)** - The pyramid volume must be calculated using \(\frac{1}{3}\).
2. **The space inside greenhouse 1 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{3} \times 12 \times 12 \times 8 \).**
**(True)** - This is the correct equation for Greenhouse 1.
3. **The space inside greenhouse 2 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{2} \times 12 \times 12 \times 8 \).**
**(False)** - The triangular prism’s volume is wrongly described (it should not use \(\frac{1}{2}\)).
4. **The space inside greenhouse 2 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{3} \times 12 \times 12 \times 8 \).**
**(False)** - The triangular prism’s volume is wrongly described (it should not use \(\frac{1}{3}\)).
5. **The space inside greenhouse 1 is 192 meters cubed greater than the space inside greenhouse 2.**
**(False)** - Volume of Greenhouse 1 = 1824 m³; volume of Greenhouse 2 = 2016 m³, thus Greenhouse 2 is greater.
### Conclusion:
The two correct statements are:
- **The space inside greenhouse 1 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{3} \times 12 \times 12 \times 8 \).**
### Greenhouse 1:
- The base: rectangular prism with dimensions 12m x 12m x 10m
- The top: square pyramid with a base of 12m x 12m and height of 8m
The volume \( V \) of the rectangular prism can be calculated as:
\[
V_{rectangular \, prism} = \text{length} \times \text{width} \times \text{height} = 12 \times 12 \times 10 = 1440 \, \text{m}^3
\]
The volume \( V \) of the square pyramid can be calculated as:
\[
V_{pyramid} = \frac{1}{3} \times \text{base area} \times \text{height} = \frac{1}{3} \times (12 \times 12) \times 8 = \frac{1}{3} \times 144 \times 8 = \frac{1152}{3} = 384 \, \text{m}^3
\]
The total volume of Greenhouse 1:
\[
V_{greenhouse 1} = V_{rectangular \, prism} + V_{pyramid} = 1440 + 384 = 1824 \, \text{m}^3
\]
### Greenhouse 2:
- The base: rectangular prism with dimensions 12m x 12m x 10m
- The top: triangular prism with a triangular base of height 8m and base of 12m, and prism height of 12m
The volume \( V \) of the triangular base can be calculated as:
\[
\text{Area of the triangular base} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 8 = 48 \, \text{m}^2
\]
The volume \( V \) of the triangular prism is:
\[
V_{triangular \, prism} = \text{base area} \times \text{height} = 48 \times 12 = 576 \, \text{m}^3
\]
The total volume of Greenhouse 2:
\[
V_{greenhouse 2} = V_{rectangular \, prism} + V_{triangular \, prism} = 1440 + 576 = 2016 \, \text{m}^3
\]
### Evaluating the Statements:
Now let's analyze the provided statements:
1. **The space inside greenhouse 1 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{2} \times 12 \times 12 \times 8 \).**
**(False)** - The pyramid volume must be calculated using \(\frac{1}{3}\).
2. **The space inside greenhouse 1 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{3} \times 12 \times 12 \times 8 \).**
**(True)** - This is the correct equation for Greenhouse 1.
3. **The space inside greenhouse 2 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{2} \times 12 \times 12 \times 8 \).**
**(False)** - The triangular prism’s volume is wrongly described (it should not use \(\frac{1}{2}\)).
4. **The space inside greenhouse 2 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{3} \times 12 \times 12 \times 8 \).**
**(False)** - The triangular prism’s volume is wrongly described (it should not use \(\frac{1}{3}\)).
5. **The space inside greenhouse 1 is 192 meters cubed greater than the space inside greenhouse 2.**
**(False)** - Volume of Greenhouse 1 = 1824 m³; volume of Greenhouse 2 = 2016 m³, thus Greenhouse 2 is greater.
### Conclusion:
The two correct statements are:
- **The space inside greenhouse 1 can be found using the equation \( V = 12 \times 12 \times 10 + \frac{1}{3} \times 12 \times 12 \times 8 \).**