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)
The volume of a rectangular pyramid is given by the formula:
Volume = (1/3) * base area * height
First, we need to find the area of the base, which is a rectangle. The formula for the area of a rectangle is:
Area = length * width
Plugging in the values we have:
Area = 9 cm * 4 cm
Area = 36 cm^2
Now, we can calculate the volume of the wedge:
Volume = (1/3) * 36 cm^2 * 21 cm
Volume = (1/3) * 756 cm^3
Volume = 252 cm^3
Therefore, the volume of the piece of cheese is 252 cubic centimeters.
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)
Since the base and height of the rectangular pyramid are congruent to the rectangular prism, we can assume that these shapes are similar. This means that their corresponding dimensions are proportional.
The volume of a rectangular prism is given by:
Volume_prism = length_prism * width_prism * height_prism
Given that the volume of the prism is 27 in^3, we can express the dimensions of the prism as:
length_prism = x
width_prism = x
height_prism = 27 / x^2
Now, for a rectangular pyramid with congruent base and height:
Volume_pyramid = (1/3) * base area_pyramid * height_pyramid
Since the base of the pyramid is a square with side length x (congruent to the prism), the base area of the pyramid is x^2:
Volume_pyramid = (1/3) * x^2 * (27 / x^2)
Volume_pyramid = 9 in^3
Therefore, the volume of the rectangular pyramid is 9 in^3.
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)
To determine which shape has the larger volume between the triangular pyramid and the triangular prism, we need to calculate the volume of each shape.
For the triangular prism:
Volume_prism = base area * height
The base of the prism is a triangle with sides of 3 inches and 5 inches. To find the area of a triangle given the side lengths, we can use Heron's formula:
s = (a + b + c) / 2
Area = sqrt(s * (s - a) * (s - b) * (s - c)), where a, b, and c are the side lengths of the triangle and s is the semiperimeter.
Calculating the area of the triangular base of the prism:
s = (3 + 5 + 8.41) / 2 = 8.705
Area_prism_base = sqrt(8.705 * (8.705 - 3) * (8.705 - 5) * (8.705 - 8.41))
Area_prism_base ≈ 11.78 in^2
Now calculate the volume of the prism:
Volume_prism = 11.78 in^2 * 8.41 inches
Volume_prism ≈ 99.2438 in^3
For the triangular pyramid:
Volume_pyramid = (1/3) * base area * height
The base of the pyramid is the same as that of the prism, so the base area is also 11.78 in^2. The height is given as 8.41 inches.
Volume_pyramid = (1/3) * 11.78 in^2 * 8.41 inches
Volume_pyramid ≈ 31.10 in^3
The larger volume is the volume of the prism. The prism is approximately 68.1338 in^3 larger than the pyramid.
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 pyramid has a larger volume by three times.
The pyramid has a larger volume by three times.
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 prism has a larger volume by 13 times.
The prism has a larger volume by Start Fraction 1 over 3 End Fraction times.
The pyramid has a larger volume by three times.
To calculate the volume of a pyramid, we use the formula:
Volume = (1/3) * base area * height
For the triangular prism:
Base area = 3 inches * 5 inches / 2 = 7.5 square inches
Height = 8.41 inches
Volume of prism = 7.5 sq in * 8.41 in = 63.075 in^3
For the triangular pyramid:
Base area = 7.5 sq in
Height = 8.41 inches
Volume of pyramid = (1/3) * 7.5 sq in * 8.41 in = 21.03 in^3
The pyramid has a larger volume of 3 times (63.075 in^3 / 21.03 in^3 = 3).
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
400 ft.3
400 ft. cubed
200 ft.3
200 ft. cubed
1,800 ft.3
1,800 ft. cubed
Given that the prism and the pyramid have the same height and the same triangular base with dimensions given, we can determine the volume of the pyramid.
The volume of a prism is given by the formula:
Volume_prism = base area * height
The base area of the prism is the area of the triangle, which is 1/2 * base * height:
Base area = 0.5 * 20 ft * 6 ft = 60 sq ft
Given that the volume of the prism is 600 ft^3, and the height is the same for both shapes, we can calculate the volume of the pyramid using the same height and triangular base:
Volume_pyramid = (1/3) * base area * height
Volume_pyramid = (1/3) * 60 sq ft * H = 600 ft^3
20 ft * H = 600 ft^3
H = 600 ft^3 / 20 ft = 30 ft
Now, we can calculate the volume of the pyramid:
Volume_pyramid = (1/3) * 60 sq ft * 30 ft = 600 ft^3
Therefore, the volume of the pyramid is 600 ft^3.