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An illustration shows a cube placed on the top and center of a larger rectangular prism. The length and width of the prism is 6 inches respectively and 2 inches high. The sides of the cube are 2 inches.
What is the volume of the composite figure?
The volume of the rectangular prism is given by the formula: length x width x height = 6 x 6 x 2 = 72 cubic inches.
The volume of the cube is given by the formula: side length x side length x side length = 2 x 2 x 2 = 8 cubic inches.
Since the cube is on top of the rectangular prism, the volume of the composite figure is the sum of the volumes of the rectangular prism and the cube.
Composite volume = 72 + 8 = 80 cubic inches.
Therefore, the volume of the composite figure is 80 cubic inches.
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An illustration shows two cubes joined on both sides of a cuboid. The sides of the cube are 2 yards. The length of the cuboid is 4.5 yards.
What is the volume of the composite figure?
First, let's calculate the volume of one cube. The volume of a cube is given by the formula: side length x side length x side length.
For one cube: 2 yards x 2 yards x 2 yards = 8 cubic yards.
Since there are two cubes, the total volume of the cubes is 8 cubic yards x 2 = 16 cubic yards.
Now, let's calculate the volume of the cuboid. The volume of a cuboid is given by the formula: length x width x height.
For the cuboid: 4.5 yards x 2 yards x 2 yards = 18 cubic yards.
Now, to find the volume of the composite figure, we add the volume of the two cubes and the volume of the cuboid.
Composite volume = 16 cubic yards (cubes) + 18 cubic yards (cuboid) = 34 cubic yards.
Therefore, the volume of the composite figure is 34 cubic yards.
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An illustration shows a hexagonal prism with the label Choosy Cheesy on the top surface. The height of the hexagonal prism is 8 centimeters. The two longer parallel sides of the hexagonal prism each measure 30 centimeters in length. One end of the prism is marked with a dashed line measuring 15 centimeters across, joining the two opposite vertices of the parallel edges, forming a triangular prism with a perpendicular measure of 8 centimeters.
A sample of gourmet cheeses came in this box. What is the volume of the box?
To calculate the volume of the box which is essentially the hexagonal prism and the triangular prism combined, we need to calculate the volume of each separately and then add them together.
First, let's calculate the volume of the hexagonal prism. The formula to find the volume of a hexagonal prism is the area of the hexagon base multiplied by the height of the prism.
Since the two longer parallel sides each measure 30 centimeters, the base of the hexagon is formed by two equilateral triangles. The area of one equilateral triangle is given by (1/2) x base x height. Using trigonometry, we can find the height of the equilateral triangle:
height = sqrt(30^2 - (15^2 / 4)) = sqrt(900 - 56.25) = sqrt(843.75) ≈ 29.05 cm
Therefore, the area of one equilateral triangle is (1/2) x 30 cm x 29.05 cm ≈ 435.75 cm². The area of the hexagon base would be twice this value.
Area of hexagon base = 2 x 435.75 cm² = 871.5 cm²
Volume of hexagonal prism = 871.5 cm² x 8 cm = 6972 cm³
Next, let's calculate the volume of the triangular prism. The volume of a triangular prism is given by the formula: (1/2) x base x height x length.
The base of the triangular prism is 15 cm, height is 8 cm, and length (the perpendicular height) is 8 cm.
Volume of triangular prism = (1/2) x 15 cm x 8 cm x 8 cm = 480 cm³
Finally, to find the total volume of the box (hexagonal prism + triangular prism):
Total volume = Volume of hexagonal prism + Volume of triangular prism
Total volume = 6972 cm³ + 480 cm³ = 7452 cm³
Therefore, the volume of the box is 7452 cubic centimeters.
wrong
I apologize for the error in the previous response. Let's correct this calculation.
To find the volume of the hexagonal prism, we need to find the area of the hexagonal base. The hexagon can be divided into six equal equilateral triangles. The formula for the area of an equilateral triangle is (sqrt(3) / 4) x side length^2.
The side length of the hexagon can be calculated using trigonometry as follows: side length = 2 x (15 / 2) / tan(30 degrees) = 15 / tan(30 degrees) = 15 / sqrt(3) ≈ 8.66 cm
The area of one equilateral triangle (and therefore one side of the hexagon) is (sqrt(3) / 4) x 8.66^2 ≈ 31.63 cm².
The area of the hexagon base is 6 x 31.63 cm² = 189.78 cm².
Now, to find the volume of the hexagonal prism:
Volume of hexagonal prism = 189.78 cm² x 8 cm = 1518.24 cm³
Next, let's calculate the volume of the triangular prism. The base of the triangular prism is an isosceles triangle with a base of 15 cm and height of 8 cm. The area of this triangle is (1/2) x base x height = (1/2) x 15 cm x 8 cm = 60 cm².
The volume of the triangular prism is this area multiplied by the length of the prism (also 8 cm):
Volume of triangular prism = 60 cm² x 8 cm = 480 cm³
To find the total volume of the box (hexagonal prism + triangular prism):
Total volume = Volume of hexagonal prism + Volume of triangular prism
Total volume = 1518.24 cm³ + 480 cm³ = 1998.24 cm³
Therefore, the correct volume of the box is 1998.24 cubic centimeters.
wrong ;-;
I apologize for the errors in the previous responses. Let's correct the calculation.
To find the volume of the box, we need to calculate the volume of the hexagonal prism and the volume of the triangular prism and then add them together.
1. Volume of the hexagonal prism:
The formula for the volume of a prism is base area x height.
The hexagon base of the prism is divided into six equilateral triangles. The area of each equilateral triangle can be found using trigonometry:
Area of equilateral triangle = (1/2) x side length^2 x sin(60 degrees)
Calculating the side length: side length = 15 cm, so each triangle has an area of (1/2) x 15^2 x sin(60) = 108.75 cm²
Since there are six triangles, the total area of the hexagon base is 6 x 108.75 = 652.5 cm²
Volume of hexagonal prism = base area x height = 652.5 cm² x 8 cm = 5220 cubic centimeters
2. Volume of the triangular prism:
The base of the triangular prism is an isosceles triangle with base 15 cm and height 8 cm. The area of this isosceles triangle is (1/2) x base x height = (1/2) x 15 cm x 8 cm = 60 cm²
Volume of triangular prism = base area x height = 60 cm² x 8 cm = 480 cubic centimeters
Finally, to find the total volume of the box:
Total volume = Volume of hexagonal prism + Volume of triangular prism
Total volume = 5220 cm³ + 480 cm³ = 5700 cm³
Therefore, the correct volume of the box is 5700 cubic centimeters. Thank you for your patience.
you got it wrong again
it's okay, you got thisUse the image to answer the question.
An illustration shows a hexagonal prism with the label Choosy Cheesy on the top surface. The height of the hexagonal prism is 8 centimeters. The two longer parallel sides of the hexagonal prism each measure 30 centimeters in length. One end of the prism is marked with a dashed line measuring 15 centimeters across, joining the two opposite vertices of the parallel edges, forming a triangular prism with a perpendicular measure of 8 centimeters.
What is the volume of the box?