Question
Alicia has molds to make ice cubes in the shapes of two pyramids that have the same height but different bases. They are shown below. Alicia says the volume of ice in shape 1, the rectangular pyramid, is three times the volume of ice in shape 2, the triangular pyramid. Is Alicia correct?
Use the formula below to determine the answer and use the drop-down menus to explain.
V = }Bh
Ice Molds
5 cm
→
5 cm
- 3 cm
3 cm
4 cm
Shape 1
4 cm
Shape 2Click the arrows to choose an answer from each menu.
The volume of ice in shape 1 is Choose...
The volume of ice in shape 2 is
Choose...
Shape 1 contains
Choose...
Alicia
Choose...
correct.
cubic centimeters.
cubic centimeters.
ice as shape 2.
Use the formula below to determine the answer and use the drop-down menus to explain.
V = }Bh
Ice Molds
5 cm
→
5 cm
- 3 cm
3 cm
4 cm
Shape 1
4 cm
Shape 2Click the arrows to choose an answer from each menu.
The volume of ice in shape 1 is Choose...
The volume of ice in shape 2 is
Choose...
Shape 1 contains
Choose...
Alicia
Choose...
correct.
cubic centimeters.
cubic centimeters.
ice as shape 2.
Answers
GPT-4o mini
To solve the problem, we need to calculate the volume of both the rectangular pyramid (shape 1) and the triangular pyramid (shape 2) using the formula for the volume of a pyramid, which is given by:
\[ V = \frac{1}{3} B h \]
where \( B \) is the area of the base, and \( h \) is the height of the pyramid.
### Shape 1: Rectangular Pyramid
1. **Base Area (B)**:
- The base is a rectangle with dimensions 5 cm and 4 cm.
- Area of the base:
\[ B = 5 \, \text{cm} \times 4 \, \text{cm} = 20 \, \text{cm}^2 \]
2. **Height (h)**:
- The height of the pyramid is 3 cm.
3. **Volume (V)**:
\[ V = \frac{1}{3} \times 20 \, \text{cm}^2 \times 3 \, \text{cm} = \frac{60}{3} = 20 \, \text{cm}^3 \]
### Shape 2: Triangular Pyramid
1. **Base Area (B)**:
- The base is a triangle with a base of 4 cm and a height of 3 cm.
- Area of the base:
\[ B = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \, \text{cm} \times 3 \, \text{cm} = \frac{12}{2} = 6 \, \text{cm}^2 \]
2. **Height (h)**:
- The height of the pyramid is also 3 cm.
3. **Volume (V)**:
\[ V = \frac{1}{3} \times 6 \, \text{cm}^2 \times 3 \, \text{cm} = \frac{18}{3} = 6 \, \text{cm}^3 \]
### Conclusion:
- **Volume of Shape 1 (Rectangular Pyramid)**: 20 cubic centimeters.
- **Volume of Shape 2 (Triangular Pyramid)**: 6 cubic centimeters.
Now we can compare the volumes:
- Volume of Shape 1 = 20 cm³
- Volume of Shape 2 = 6 cm³
To check Alicia's assertion:
- \( 3 \times \text{Volume of shape 2} = 3 \times 6 \, \text{cm}^3 = 18 \, \text{cm}^3 \)
- Since 20 cm³ (volume of shape 1) is not equal to 18 cm³, Alicia is not correct.
### Fill in the drop-down menus:
1. The volume of ice in shape 1 is **20 cubic centimeters.**
2. The volume of ice in shape 2 is **6 cubic centimeters.**
3. Shape 1 contains **more ice than** shape 2.
4. Alicia is **not** correct.
\[ V = \frac{1}{3} B h \]
where \( B \) is the area of the base, and \( h \) is the height of the pyramid.
### Shape 1: Rectangular Pyramid
1. **Base Area (B)**:
- The base is a rectangle with dimensions 5 cm and 4 cm.
- Area of the base:
\[ B = 5 \, \text{cm} \times 4 \, \text{cm} = 20 \, \text{cm}^2 \]
2. **Height (h)**:
- The height of the pyramid is 3 cm.
3. **Volume (V)**:
\[ V = \frac{1}{3} \times 20 \, \text{cm}^2 \times 3 \, \text{cm} = \frac{60}{3} = 20 \, \text{cm}^3 \]
### Shape 2: Triangular Pyramid
1. **Base Area (B)**:
- The base is a triangle with a base of 4 cm and a height of 3 cm.
- Area of the base:
\[ B = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \, \text{cm} \times 3 \, \text{cm} = \frac{12}{2} = 6 \, \text{cm}^2 \]
2. **Height (h)**:
- The height of the pyramid is also 3 cm.
3. **Volume (V)**:
\[ V = \frac{1}{3} \times 6 \, \text{cm}^2 \times 3 \, \text{cm} = \frac{18}{3} = 6 \, \text{cm}^3 \]
### Conclusion:
- **Volume of Shape 1 (Rectangular Pyramid)**: 20 cubic centimeters.
- **Volume of Shape 2 (Triangular Pyramid)**: 6 cubic centimeters.
Now we can compare the volumes:
- Volume of Shape 1 = 20 cm³
- Volume of Shape 2 = 6 cm³
To check Alicia's assertion:
- \( 3 \times \text{Volume of shape 2} = 3 \times 6 \, \text{cm}^3 = 18 \, \text{cm}^3 \)
- Since 20 cm³ (volume of shape 1) is not equal to 18 cm³, Alicia is not correct.
### Fill in the drop-down menus:
1. The volume of ice in shape 1 is **20 cubic centimeters.**
2. The volume of ice in shape 2 is **6 cubic centimeters.**
3. Shape 1 contains **more ice than** shape 2.
4. Alicia is **not** correct.