What is the surface area of the rectangular pyramid?
An illustration shows a pyramid with a rectangular base. The length of the base is 8 millimeters and the width 10 millimeters. The perpendicular heights from the apex to the base length is 12 millimeters and to the base width is 15 millimeters. The perpendicular heights are denoted by two right angle symbols on two of the lateral faces from the apex to the base by dashed lines.
To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of each of the four triangular faces.
First, let's calculate the area of the base:
Area of the base = length x width
Area of the base = 8 mm x 10 mm
Area of the base = 80 mm²
Next, let's calculate the area of each of the four triangular faces. Since the pyramid is rectangular, the triangles are all congruent and we only need to calculate the area of one of them:
Area of one triangular face = (1/2) x base x height
Area of one triangular face = (1/2) x 8 mm x 12 mm
Area of one triangular face = 48 mm²
Now, we can find the total surface area of the pyramid by adding the area of the base and the areas of the four triangular faces:
Total surface area = Area of base + (4 x Area of one triangular face)
Total surface area = 80 mm² + (4 x 48 mm²)
Total surface area = 80 mm² + 192 mm²
Total surface area = 272 mm²
Therefore, the surface area of the rectangular pyramid is 272 square millimeters.
wrong
Apologies for the mistake. Let's correct the calculation:
The total surface area of a rectangular pyramid can be calculated using the formula:
Surface Area = Base Area + (0.5 x Perimeter of Base x Slant Height)
First, let's calculate the perimeter of the base:
Perimeter of Base = 2(length + width)
Perimeter of Base = 2(8 mm + 10 mm)
Perimeter of Base = 2(18 mm)
Perimeter of Base = 36 mm
Next, let's calculate the slant height of the triangular faces. We can use the Pythagorean theorem to find the slant height:
Slant Height = sqrt(base height^2 + (0.5 x Perimeter of Base)^2)
Slant Height = sqrt(12 mm^2 + 18 mm^2)
Slant Height = sqrt(144 mm^2 + 324 mm^2)
Slant Height = sqrt(468 mm^2)
Slant Height ≈ 21.63 mm
Now, we can calculate the surface area:
Surface Area = Base Area + (0.5 x Perimeter of Base x Slant Height)
Surface Area = (8 mm x 10 mm) + (0.5 x 36 mm x 21.63 mm)
Surface Area = 80 mm^2 + (0.5 x 777.18 mm^2)
Surface Area = 80 mm^2 + 388.59 mm^2
Surface Area = 468.59 mm^2
Therefore, the correct surface area of the rectangular pyramid is approximately 468.59 square millimeters.
An illustration shows a pyramid with a square base. The side of the base is labeled 15 feet. The perpendicular height denoted by a right angle symbol on the lateral face from the apex to the base is labeled 9.9 feet.
Solve for the surface area of the square pyramid.
To find the surface area of the square pyramid, we need to calculate the area of the base and the area of each of the four triangular faces.
First, let's calculate the area of the base:
Area of the base = side x side
Area of the base = 15 feet x 15 feet
Area of the base = 225 square feet
Next, let's calculate the area of each of the four triangular faces. Since the pyramid is square-based, the triangles are all congruent and we only need to calculate the area of one of them:
Area of one triangular face = (1/2) x base x height
Area of one triangular face = (1/2) x 15 feet x 9.9 feet
Area of one triangular face = 74.25 square feet
Now, we can find the total surface area of the pyramid by adding the area of the base and the areas of the four triangular faces:
Total surface area = Area of base + (4 x Area of one triangular face)
Total surface area = 225 square feet + (4 x 74.25 square feet)
Total surface area = 225 square feet + 297 square feet
Total surface area = 522 square feet
Therefore, the surface area of the square pyramid is 522 square feet.
factory makes nylon tea bags. The bags are in the shape of a triangular pyramid. How much nylon is needed to make 50 tea bags given the following dimensions?
Base of the equilateral triangles: 40 mm
Height of the base triangle: 35 mm
Slant height of the equilateral triangular faces: 45 mm
To find the amount of nylon needed to make 50 tea bags in the shape of a triangular pyramid, we need to calculate the total surface area of one pyramid and then multiply it by the number of tea bags.
First, let's calculate the surface area of one triangular pyramid. Since the base triangle is equilateral, we can treat it as an equilateral triangle.
The formula for the surface area of a triangular pyramid is:
Surface Area = Base Area + (0.5 x Perimeter of Base x Slant Height)
1. Calculate the base area (equilateral triangle):
Base Area = (sqrt(3) / 4) x side^2
Base Area = (sqrt(3) / 4) x 40 mm^2
Base Area ≈ 692.82 mm²
2. Calculate the perimeter of the base (equilateral triangle):
Perimeter of Base = 3 x side
Perimeter of Base = 3 x 40 mm
Perimeter of Base = 120 mm
3. Calculate the surface area of the triangular faces:
Surface Area = 692.82 mm² + (0.5 x 120 mm x 45 mm)
Surface Area = 692.82 mm² + 2700 mm²
Surface Area = 3392.82 mm²
Now, to find the total amount of nylon needed for 50 tea bags:
Total Nylon Needed = Surface Area x 50 tea bags
Total Nylon Needed = 3392.82 mm² x 50
Total Nylon Needed = 169,641 mm²
Therefore, 169,641 square millimeters of nylon are needed to make 50 tea bags in the shape of a triangular pyramid.
answer options:
153,125 mm2
153,125 mm squared
3,400 mm2
3,400 mm squared
3,062.5 mm2
3,062.5 mm squared
170,000 mm2
To calculate the surface area of one triangular pyramid and then find the total amount of nylon needed to make 50 tea bags, we follow these steps:
Given dimensions:
Base of the equilateral triangles (Side length): 40 mm
Height of the base triangle: 35 mm
Slant height of the equilateral triangular faces: 45 mm
1. Calculate the base area of the equilateral triangle:
Base Area = (sqrt(3) / 4) x side^2
Base Area = (sqrt(3) / 4) x 40 mm^2
Base Area ≈ 692.82 mm²
2. Calculate the lateral surface area of the pyramid (sum of all triangular faces):
Lateral Surface Area = Perimeter of Base x Slant Height / 2
Perimeter of Base = 3 x side = 3 x 40 mm = 120 mm
Lateral Surface Area = 120 mm x 45 mm / 2 = 2700 mm²
3. Total surface area of the pyramid:
Total Surface Area = Base Area + Lateral Surface Area
Total Surface Area = 692.82 mm² + 2700 mm² = 3392.82 mm²
4. Total nylon needed for 50 tea bags:
Total Nylon Needed = Total Surface Area x 50 bags
Total Nylon Needed = 3392.82 mm² x 50 = 169,641 mm²
Among the given options, the closest answer to 169,641 mm² is 170,000 mm². Therefore, the answer is:
170,000 mm²