Question

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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)

Answer 1

[AI-generated answer — Bot GPT 3.5]

The volume of a rectangular pyramid is given by the formula V = (1/3) * B * h, where B is the area of the base and h is the height of the pyramid.

Since the volume of the pyramid is 480 in^3, we have 480 = (1/3) * B * h.

We also know that the base of the pyramid is a rectangle, so B = length * width.

Since the base and height of the prism are congruent to the pyramid, we have the same base and height measurements for both shapes.

Let the length and width of the base of the pyramid be l and w, respectively. Let the height of both the prism and pyramid be h.

Therefore, the volume of the prism V' = l * w * h.

Since the base and height of the prism are congruent to the pyramid, we have l = w and h = h.

Substitute l = w and h = h into the volume formula for the prism:
V' = (l * l) * h
V' = l^2 * h

Since the volume of the pyramid is 480 in^3, we have:
480 = (1/3) * l * w * h

Substitute l = w and h = h into the volume formula for the pyramid:
480 = (1/3) * l^2 * h
l^2 * h = 1440
l^2 * h = V'

Therefore, the volume of the prism is 1440 in^3.

Answer 2

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

600 ft.3
600 ft. cubed

200 ft.3
200 ft. cubed

400 ft.3
400 ft. cubed

1,800 ft.3

Answer 3

[AI-generated answer — Bot GPT 3.5]

The volume of a prism with a triangular base is given by the formula V = (1/2) * base * height * h, where the base and height refer to the dimensions of the triangle and h is the height of the prism.

Given that the volume of the prism is 600 ft^3, we have:
600 = (1/2) * 20 * 6 * h
600 = 60 * h
h = 10 ft

Now, the volume of a pyramid with a triangular base is given by the formula V = (1/3) * base * height * h.

Since the base and height dimensions are the same for both the prism and the pyramid, the volume of the pyramid would be:
V' = (1/3) * 20 * 6 * 10
V' = (1/3) * 120 * 10
V' = 400 ft^3

Therefore, the volume of the pyramid is 400 ft^3.

Answer 4

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?

(1 point)
in.3

Answer 5

[AI-generated answer — Bot GPT 3.5]

To find the volume of the composite figure consisting of a rectangular prism and a cube, we need to find the volumes of each shape separately and then sum them up.

Volume of the rectangular prism:
Volume = Length * Width * Height
Volume = 11 * 3 * 4 = 132 in^3

Volume of the cube:
Since all sides of the cube are 3 inches, the volume of the cube is the side cubed:
Volume = Side^3 = 3^3 = 27 in^3

Total volume of the composite figure:
Total Volume = Volume of prism + Volume of cube
Total Volume = 132 + 27 = 159 in^3

Therefore, the volume of the composite figure is 159 in^3.

Answer 6

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?

(1 point)

Answer 7

[AI-generated answer — Bot GPT 3.5]

To find the volume of the composite figure made up of three rectangular prisms, we need to calculate the volumes of each prism separately and then sum them up.

Volume of the side prisms:
Volume = Length * Width * Height
Volume = 16 * 7 * 7 = 784 cubic inches
Since there are two side prisms, the total volume for both side prisms is:
Total Volume for side prisms = 2 * 784 = 1568 cubic inches

Volume of the center prism:
Volume = Length * Width * Height
Volume = 10 * 8 * 7 = 560 cubic inches

Total volume of the composite figure:
Total Volume = Volume of side prisms + Volume of center prism
Total Volume = 1568 + 560 = 2128 cubic inches

Therefore, the volume of the composite figure is 2128 cubic inches.

Answer 8

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?

Answer 9

[AI-generated answer — Bot GPT 3.5]

To find the volume of the table, we need to calculate the volume of the table top and the volume of the pedestal separately, and then add them up.

1. Volume of the table top (rectangular box):
Volume = Length * Width * Depth
Volume = 30 * 30 * 2 = 1800 cubic inches

2. Volume of the pedestal (rectangular box):
The pedestal consists of two parts: the rectangular block and the rectangular base.

a) Volume of the rectangular block:
Volume = Length * Width * Height
Volume = 8 * 8 * 25 = 1600 cubic inches

b) Volume of the rectangular base:
Volume = Length * Width * Height
Volume = 12 * 12 * 4 = 576 cubic inches

Total volume of the pedestal:
Total Volume = Volume of rectangular block + Volume of rectangular base
Total Volume = 1600 + 576 = 2176 cubic inches

Total volume of the table:
Total Volume = Volume of table top + Volume of pedestal
Total Volume = 1800 + 2176 = 3976 cubic inches

Therefore, the volume of the table is 3976 cubic inches.

Answer 10

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?

(1 point)

Answer 11

[AI-generated answer — Bot GPT 3.5]

To find the volume of the movie theater, we need to calculate the volumes of the central square prism and the four smaller square prisms separately, and then sum them up.

1. Volume of the central square prism:
Volume = side^2 * height
Volume = 44^2 * 20 = 38,720 cubic meters

2. Volume of each of the four smaller square prisms:
Volume = side^2 * height
Volume = 15^2 * 20 = 4500 cubic meters
Since there are four smaller square prisms, the total volume for all of them is:
Total Volume for smaller prisms = 4 * 4500 = 18,000 cubic meters

Total volume of the movie theater:
Total Volume = Volume of central square prism + Volume of smaller square prisms
Total Volume = 38,720 + 18,000 = 56,720 cubic meters

Therefore, the volume of the movie theater is 56,720 cubic meters.