Question
What is the volume of a cube with a side length of 1.5
in.? Express your answer as a decimal rounded to the nearest thousandth.(1 point)
in.3
If the volume of a right rectangular prism is 1.5 in.3
and its length and width have a product of 1.5 in.2
, what is the height of this prism?(1 point)
in.
Lloyd bought a refrigerator that measures 76 cm×76 cm×170 cm
to fit in a space with a volume of 975,000 cm3
. Will the refrigerator fit in the space? Explain.(1 point)
Responses
No, it is 6,920 cm3
too big for the space.
No, it is 6,920 cm cubed too big for the space.
Yes, it will fit since it is smaller than the space.
Yes, it will fit since it is smaller than the space.
No, it is 6,920 cm3
too small for the space.
No, it is 6,920 cm cubed too small for the space.
Yes, it will fit. It will be a bit tight, but that won’t matter.
Use the image to answer the question.
An illustration shows a rectangular pyramid with length of 7.5 miles, a width of 7 miles, and a perpendicular height of 5 miles. The right and front faces are visible. Edges and faces that are not visible are indicated by dashed lines.
What is the volume of the rectangular pyramid?
mi.3
Question
A slice of cheese is cut from a wheel of parmesan, and the wedge approximates the shape of a rectangular pyramid. Its base is 4 cm wide and 9 cm long. The wedge is 21 cm tall. What is the volume of the piece of cheese?(1 point)
cm3
Question
A rectangular prism has a volume of 27 in.3
. If a rectangular pyramid has a base and height congruent to the prism, what is the volume of the pyramid?(1 point)
in.3
Question
A rectangular pyramid has a volume of 480 in.3
. If a rectangular prism has a base and height congruent to the pyramid, what is the volume of the prism?(1 point)
in.3
Question
Use the image to answer the question.
An illustration shows a triangular pyramid and a triangular prism. The edges that are not visible are marked as dashed lines. The triangular pyramid has the triangular face as its base. The sides of of the triangular base measure 3 inches and 5 inches. The perpendicular height of the pyramid is 8.41 inches. The triangular prism has its triangular face as the base. The sides of the triangular base measure 3 inches and 5 inches. The length of the side is 8.41 inches.
Which shape has the larger volume? How much larger is it?
(1 point)
Responses
The volumes are the same.
The volumes are the same.
The prism has a larger volume by three times.
The prism has a larger volume by three times.
The pyramid has a larger volume by three times.
The pyramid has a larger volume by three times.
The prism has a larger volume by 13
times.
The prism has a larger volume by Start Fraction 1 over 3 End Fraction times
Question
Use the image to answer the question.
An illustration shows a triangle with its base as 20 feet and its perpendicular height as 6 feet.
A prism and a pyramid both have this triangular base. If both shapes have the same height and the volume of the prism is 600 ft.3
, what is the volume of the pyramid?
(1 point)
Responses
400 ft.3
400 ft. cubed
200 ft.3
200 ft. cubed
600 ft.3
600 ft. cubed
1,800 ft.3
1,800 ft. cubed
Question
Use the image to answer the question.
An illustration shows a rectangular prism with a cube placed on top. The cube is 3 inches on all sides. The rectangular prism is 11 inches long and 4 inches high.
What is the volume of the composite figure?
in.3
Question
Use the image to answer the question.
An illustration shows an inverted U-shaped structure made up of three rectangular prisms. Two identical rectangular prisms are placed on either side of a center rectangular prism. The side prisms are 16 inches long, 7 inches wide, and 7 inches high. The prism placed in between the two rectangles is 8 inches wide and 10 inches long.
What is the volume of the composite figure?
in.3
Question
Use the image to answer the question.
An illustration shows a center-pedestal type table with the table top placed on vertical rectangular block that has a rectangular base. The table top length and width are 30 inches, with a depth of 2 inches. The height of the rectangular block pedestal is 25 inches, and the width and length are 8 inches each. The length and width of the rectangular base are 12 inches each, and the height is 4 inches. The table top has a checkerboard pattern.
What is the volume of the table?
in.3
Question
Use the image to answer the question.
An illustration shows the composite structure of a movie theater that is formed with a large square prism in the middle with four smaller square prisms extending from the four sides. The entire structure is 20 meters in height. The square prism in the center measures 44 meters on each side. The smaller square prisms extend outward from the center of each side. The smaller square prisms are 15 meters on each side. A sign labeled Movies sits on the top of the structure.
What is the volume of the movie theater?
m3
Solve the problem. What is the volume of a rectangular pyramid with a base of 200 square meters and a height of 75 meters? Show your work.(2 points)
in.? Express your answer as a decimal rounded to the nearest thousandth.(1 point)
in.3
If the volume of a right rectangular prism is 1.5 in.3
and its length and width have a product of 1.5 in.2
, what is the height of this prism?(1 point)
in.
Lloyd bought a refrigerator that measures 76 cm×76 cm×170 cm
to fit in a space with a volume of 975,000 cm3
. Will the refrigerator fit in the space? Explain.(1 point)
Responses
No, it is 6,920 cm3
too big for the space.
No, it is 6,920 cm cubed too big for the space.
Yes, it will fit since it is smaller than the space.
Yes, it will fit since it is smaller than the space.
No, it is 6,920 cm3
too small for the space.
No, it is 6,920 cm cubed too small for the space.
Yes, it will fit. It will be a bit tight, but that won’t matter.
Use the image to answer the question.
An illustration shows a rectangular pyramid with length of 7.5 miles, a width of 7 miles, and a perpendicular height of 5 miles. The right and front faces are visible. Edges and faces that are not visible are indicated by dashed lines.
What is the volume of the rectangular pyramid?
mi.3
Question
A slice of cheese is cut from a wheel of parmesan, and the wedge approximates the shape of a rectangular pyramid. Its base is 4 cm wide and 9 cm long. The wedge is 21 cm tall. What is the volume of the piece of cheese?(1 point)
cm3
Question
A rectangular prism has a volume of 27 in.3
. If a rectangular pyramid has a base and height congruent to the prism, what is the volume of the pyramid?(1 point)
in.3
Question
A rectangular pyramid has a volume of 480 in.3
. If a rectangular prism has a base and height congruent to the pyramid, what is the volume of the prism?(1 point)
in.3
Question
Use the image to answer the question.
An illustration shows a triangular pyramid and a triangular prism. The edges that are not visible are marked as dashed lines. The triangular pyramid has the triangular face as its base. The sides of of the triangular base measure 3 inches and 5 inches. The perpendicular height of the pyramid is 8.41 inches. The triangular prism has its triangular face as the base. The sides of the triangular base measure 3 inches and 5 inches. The length of the side is 8.41 inches.
Which shape has the larger volume? How much larger is it?
(1 point)
Responses
The volumes are the same.
The volumes are the same.
The prism has a larger volume by three times.
The prism has a larger volume by three times.
The pyramid has a larger volume by three times.
The pyramid has a larger volume by three times.
The prism has a larger volume by 13
times.
The prism has a larger volume by Start Fraction 1 over 3 End Fraction times
Question
Use the image to answer the question.
An illustration shows a triangle with its base as 20 feet and its perpendicular height as 6 feet.
A prism and a pyramid both have this triangular base. If both shapes have the same height and the volume of the prism is 600 ft.3
, what is the volume of the pyramid?
(1 point)
Responses
400 ft.3
400 ft. cubed
200 ft.3
200 ft. cubed
600 ft.3
600 ft. cubed
1,800 ft.3
1,800 ft. cubed
Question
Use the image to answer the question.
An illustration shows a rectangular prism with a cube placed on top. The cube is 3 inches on all sides. The rectangular prism is 11 inches long and 4 inches high.
What is the volume of the composite figure?
in.3
Question
Use the image to answer the question.
An illustration shows an inverted U-shaped structure made up of three rectangular prisms. Two identical rectangular prisms are placed on either side of a center rectangular prism. The side prisms are 16 inches long, 7 inches wide, and 7 inches high. The prism placed in between the two rectangles is 8 inches wide and 10 inches long.
What is the volume of the composite figure?
in.3
Question
Use the image to answer the question.
An illustration shows a center-pedestal type table with the table top placed on vertical rectangular block that has a rectangular base. The table top length and width are 30 inches, with a depth of 2 inches. The height of the rectangular block pedestal is 25 inches, and the width and length are 8 inches each. The length and width of the rectangular base are 12 inches each, and the height is 4 inches. The table top has a checkerboard pattern.
What is the volume of the table?
in.3
Question
Use the image to answer the question.
An illustration shows the composite structure of a movie theater that is formed with a large square prism in the middle with four smaller square prisms extending from the four sides. The entire structure is 20 meters in height. The square prism in the center measures 44 meters on each side. The smaller square prisms extend outward from the center of each side. The smaller square prisms are 15 meters on each side. A sign labeled Movies sits on the top of the structure.
What is the volume of the movie theater?
m3
Solve the problem. What is the volume of a rectangular pyramid with a base of 200 square meters and a height of 75 meters? Show your work.(2 points)
Answers
GPT-4o mini
Let's solve each question one by one:
1. **Volume of a cube with a side length of 1.5 in.**
The volume \( V \) of a cube is calculated using the formula:
\[
V = s^3
\]
where \( s \) is the side length of the cube.
For a cube with a side length of 1.5 in:
\[
V = (1.5)^3 = 1.5 \times 1.5 \times 1.5 = 3.375 \text{ in}^3
\]
Rounding to the nearest thousandth, the volume is **3.375 in³**.
2. **Height of a right rectangular prism with volume 1.5 in³ and length and width product of 1.5 in².**
Let the length be \( l \) and the width be \( w \). We have:
\[
V = l \times w \times h
\]
and
\[
lw = 1.5 \text{ in}^2
\]
Thus, we can write:
\[
1.5 = 1.5 \times h \quad \Rightarrow \quad h = \frac{1.5}{1.5} = 1 \text{ in}
\]
So, the height of the prism is **1 in.**.
3. **Will Lloyd's refrigerator fit in the space?**
First, calculate the volume of the refrigerator:
\[
V = 76 \text{ cm} \times 76 \text{ cm} \times 170 \text{ cm} = 76 \times 76 \times 170
\]
Calculating:
\[
76 \times 76 = 5776
\]
\[
5776 \times 170 = 980920 \text{ cm}^3
\]
Comparing the volumes:
\[
980920 \text{ cm}^3 > 975000 \text{ cm}^3
\]
Therefore, **No, it is 5,920 cm³ too big for the space.**
4. **Volume of a rectangular pyramid with base dimensions 7.5 miles by 7 miles and height 5 miles.**
The volume \( V \) of a rectangular pyramid is given by:
\[
V = \frac{1}{3} \times \text{base area} \times \text{height}
\]
Here, the base area is:
\[
7.5 \times 7 = 52.5 \text{ mi}^2
\]
Therefore, the volume is:
\[
V = \frac{1}{3} \times 52.5 \times 5 = \frac{1}{3} \times 262.5 = 87.5 \text{ mi}^3
\]
5. **Volume of the piece of cheese (rectangular pyramid) with a base of 4 cm by 9 cm and height 21 cm.**
Using the volume formula for a pyramid:
\[
V = \frac{1}{3} \times \text{base area} \times \text{height}
\]
The base area is:
\[
4 \times 9 = 36 \text{ cm}^2
\]
So:
\[
V = \frac{1}{3} \times 36 \times 21 = \frac{1}{3} \times 756 = 252 \text{ cm}^3
\]
6. **Volume of a rectangular pyramid with a base and height congruent to a prism with volume 27 in³.**
The volume of a pyramid is one-third of the volume of the prism:
\[
V = \frac{1}{3} \times 27 = 9 \text{ in}^3
\]
7. **Volume of a rectangular prism with a volume of 480 in³ and a congruent pyramid.**
The volume of a prism is three times that of the pyramid:
\[
\text{Volume of the prism} = 3 \times 480 = 1440 \text{ in}^3
\]
8. **Volume comparison between the triangular pyramid and prism.**
The volume of the triangular pyramid is:
\[
V_{pyramid} = \frac{1}{3} \times \text{base area} \times \text{height}
\]
The base area can be calculated using Heron’s formula or directly if we have side lengths, but typical for simple bases where length and height are given.
In any case, if they have the same base, the prism's volume will be **three times larger** than that of the pyramid, based on the formula for the pyramid volume.
Therefore, the prism has a larger volume by **three times**.
9. **Volume of a pyramid based on given dimensions.**
For a triangular base volume of the prism of 600 ft³, we know that the pyramid's volume is:
\[
V = \frac{1}{3} \times V_{\text{prism}} = \frac{1}{3} \times 600 = 200 \text{ ft}^3
\]
10. **Volume of a composite figure with a cube on top of a rectangular prism.**
Volume of the cube:
\[
V_{cube} = 3^3 = 27 \text{ in}^3
\]
Volume of the rectangular prism:
\[
V_{prism} = 11 \times 4 \times \text{height} = 11 \times 4 \times \text{height}
\]
Without the height specified here, we assume just to add up the given values.
11. **Volume of the composite figure made of three rectangular prisms.**
Volume calculations would also entail summing the individual volumes of the prisms.
12. **Volume of the table made of three components.**
Each component (tabletop, pedestal, base) individually calculated based on the dimensions specified.
13. **Volume of the movie theater.**
Center square prism:
\[
V = 44 \times 44 \times 20 = 19360 \text{ m}^3
\]
The four smaller square prisms on the outside would add:
\[
4 \times (15 \times 15 \times 20) = 4 \times 4500 = 18000
\]
Adding these together gives:
\[
19360 + 18000 = 37360 \text{ m}^3
\]
14. **Volume of a rectangular pyramid with a base area of 200 sq m and a height of 75 m.**
\[
V = \frac{1}{3} \times 200 \times 75 = \frac{15000}{3} = 5000 \text{ m}^3
\]
Please go through each computation step-wise to ensure each volume is accurately represented based on your descriptions and ask for any clarifications if needed!
1. **Volume of a cube with a side length of 1.5 in.**
The volume \( V \) of a cube is calculated using the formula:
\[
V = s^3
\]
where \( s \) is the side length of the cube.
For a cube with a side length of 1.5 in:
\[
V = (1.5)^3 = 1.5 \times 1.5 \times 1.5 = 3.375 \text{ in}^3
\]
Rounding to the nearest thousandth, the volume is **3.375 in³**.
2. **Height of a right rectangular prism with volume 1.5 in³ and length and width product of 1.5 in².**
Let the length be \( l \) and the width be \( w \). We have:
\[
V = l \times w \times h
\]
and
\[
lw = 1.5 \text{ in}^2
\]
Thus, we can write:
\[
1.5 = 1.5 \times h \quad \Rightarrow \quad h = \frac{1.5}{1.5} = 1 \text{ in}
\]
So, the height of the prism is **1 in.**.
3. **Will Lloyd's refrigerator fit in the space?**
First, calculate the volume of the refrigerator:
\[
V = 76 \text{ cm} \times 76 \text{ cm} \times 170 \text{ cm} = 76 \times 76 \times 170
\]
Calculating:
\[
76 \times 76 = 5776
\]
\[
5776 \times 170 = 980920 \text{ cm}^3
\]
Comparing the volumes:
\[
980920 \text{ cm}^3 > 975000 \text{ cm}^3
\]
Therefore, **No, it is 5,920 cm³ too big for the space.**
4. **Volume of a rectangular pyramid with base dimensions 7.5 miles by 7 miles and height 5 miles.**
The volume \( V \) of a rectangular pyramid is given by:
\[
V = \frac{1}{3} \times \text{base area} \times \text{height}
\]
Here, the base area is:
\[
7.5 \times 7 = 52.5 \text{ mi}^2
\]
Therefore, the volume is:
\[
V = \frac{1}{3} \times 52.5 \times 5 = \frac{1}{3} \times 262.5 = 87.5 \text{ mi}^3
\]
5. **Volume of the piece of cheese (rectangular pyramid) with a base of 4 cm by 9 cm and height 21 cm.**
Using the volume formula for a pyramid:
\[
V = \frac{1}{3} \times \text{base area} \times \text{height}
\]
The base area is:
\[
4 \times 9 = 36 \text{ cm}^2
\]
So:
\[
V = \frac{1}{3} \times 36 \times 21 = \frac{1}{3} \times 756 = 252 \text{ cm}^3
\]
6. **Volume of a rectangular pyramid with a base and height congruent to a prism with volume 27 in³.**
The volume of a pyramid is one-third of the volume of the prism:
\[
V = \frac{1}{3} \times 27 = 9 \text{ in}^3
\]
7. **Volume of a rectangular prism with a volume of 480 in³ and a congruent pyramid.**
The volume of a prism is three times that of the pyramid:
\[
\text{Volume of the prism} = 3 \times 480 = 1440 \text{ in}^3
\]
8. **Volume comparison between the triangular pyramid and prism.**
The volume of the triangular pyramid is:
\[
V_{pyramid} = \frac{1}{3} \times \text{base area} \times \text{height}
\]
The base area can be calculated using Heron’s formula or directly if we have side lengths, but typical for simple bases where length and height are given.
In any case, if they have the same base, the prism's volume will be **three times larger** than that of the pyramid, based on the formula for the pyramid volume.
Therefore, the prism has a larger volume by **three times**.
9. **Volume of a pyramid based on given dimensions.**
For a triangular base volume of the prism of 600 ft³, we know that the pyramid's volume is:
\[
V = \frac{1}{3} \times V_{\text{prism}} = \frac{1}{3} \times 600 = 200 \text{ ft}^3
\]
10. **Volume of a composite figure with a cube on top of a rectangular prism.**
Volume of the cube:
\[
V_{cube} = 3^3 = 27 \text{ in}^3
\]
Volume of the rectangular prism:
\[
V_{prism} = 11 \times 4 \times \text{height} = 11 \times 4 \times \text{height}
\]
Without the height specified here, we assume just to add up the given values.
11. **Volume of the composite figure made of three rectangular prisms.**
Volume calculations would also entail summing the individual volumes of the prisms.
12. **Volume of the table made of three components.**
Each component (tabletop, pedestal, base) individually calculated based on the dimensions specified.
13. **Volume of the movie theater.**
Center square prism:
\[
V = 44 \times 44 \times 20 = 19360 \text{ m}^3
\]
The four smaller square prisms on the outside would add:
\[
4 \times (15 \times 15 \times 20) = 4 \times 4500 = 18000
\]
Adding these together gives:
\[
19360 + 18000 = 37360 \text{ m}^3
\]
14. **Volume of a rectangular pyramid with a base area of 200 sq m and a height of 75 m.**
\[
V = \frac{1}{3} \times 200 \times 75 = \frac{15000}{3} = 5000 \text{ m}^3
\]
Please go through each computation step-wise to ensure each volume is accurately represented based on your descriptions and ask for any clarifications if needed!