A welder decides to make a pyramid shaped paperweight out of flat steel. the height of the pyamid is to be 3.5in and the square base has edges of 2.0in. how much sheet metal, in square inches, will he need to form the shell? to give weight, he is going to fill it with sand. how many in3 will need to fill the paperweight?
To calculate the amount of sheet metal needed to form the shell of the pyramid-shaped paperweight, we need to find the area of the four triangular sides and the area of the base.
Step 1: Calculate the area of the base:
The base of the pyramid is square-shaped, with each side measuring 2.0 inches. Therefore, the area of the base can be calculated as:
Area of base = side length^2 = 2.0 inches * 2.0 inches = 4.0 square inches.
Step 2: Calculate the area of the triangular sides:
The pyramid has four triangular sides that are all congruent. To calculate the area of one triangular side, we need to find the base and the height of the triangle. The base of each triangle is the length of the side of the base, which is 2.0 inches. The height of the triangle is the height of the pyramid, which is 3.5 inches.
Area of one triangular side = (base * height) / 2 = (2.0 inches * 3.5 inches) / 2 = 7.0 square inches.
Since there are four triangular sides, the total area of the triangular sides is:
Total area of triangular sides = 4 * (Area of one triangular side) = 4 * 7.0 square inches = 28.0 square inches.
Step 3: Calculate the total area of the shell:
The total area of the shell is the sum of the area of the base and the total area of the triangular sides.
Total area of the shell = Area of base + Total area of triangular sides = 4.0 square inches + 28.0 square inches = 32.0 square inches.
Therefore, the welder will need approximately 32.0 square inches of sheet metal to form the shell of the pyramid-shaped paperweight.
To calculate the volume of sand needed to fill the pyramid-shaped paperweight, we need to find the volume of the pyramid.
Step 4: Calculate the volume of the pyramid:
The volume of a pyramid can be calculated using the formula:
Volume of pyramid = (base area * height) / 3.
Since the base area of our pyramid is already calculated as 4.0 square inches, and the height of the pyramid is given as 3.5 inches, we can substitute these values into the formula.
Volume of pyramid = (4.0 square inches * 3.5 inches) / 3 = 14.0 cubic inches.
Therefore, the welder will need approximately 14.0 cubic inches of sand to fill the paperweight.
To find the surface area of the pyramid-shaped paperweight, we need to calculate the area of each face and then add them up. Since all the faces of a pyramid are triangles, we can calculate the area of one triangle face and multiply it by the number of faces.
First, let's calculate the area of one triangular face:
The base of the triangle is one of the edges of the square base, which is 2.0 inches.
The height of the triangle is the height of the pyramid, which is 3.5 inches.
The formula for the area of a triangle is: (base * height) / 2.
Area of one triangular face = (2.0 inches * 3.5 inches) / 2 = 3.5 square inches.
Since there are four triangular faces on a pyramid, the total surface area of the pyramid is: 4 * 3.5 square inches = 14 square inches.
Therefore, the welder will need approximately 14 square inches of sheet metal to form the shell of the pyramid-shaped paperweight.
To calculate the volume of the pyramid-shaped paperweight, we need to calculate the volume of a pyramid and subtract the volume of the hollow space inside.
The formula for the volume of a pyramid is: (base * height) / 3.
Volume of the pyramid = (2.0 inches * 2.0 inches * 3.5 inches) / 3 = 4.67 cubic inches.
Since the welder plans to fill the paperweight with sand, we need to subtract the volume of the hollow space inside. In this case, the hollow space is a tetrahedron, which is one-fourth the volume of the whole pyramid.
Volume of the hollow space = (4.67 cubic inches) / 4 = 1.17 cubic inches.
Therefore, the welder will need approximately 4.67 - 1.17 = 3.5 cubic inches of sand to fill the paperweight.