Use the image to answer the question.
An illustration shows a pyramid with a rectangular base. The length of the base is labeled 20 centimeters and the width as 18 centimeters. The perpendicular heights from the apex to the base length of 20 is 16.6 centimeters and to the base width of 18 is 17.2 centimeters. 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.
What is the surface of the rectangular pyramid?
(1 point)
Responses
1,178.8 cm squared
102,787.2 cm squared
1,001.6 cm squared
1,643.2 cm squared
To calculate the surface area of a rectangular pyramid, you need to find the sum of the areas of its base and its four triangular faces.
First, calculate the area of the base:
Area of the base = length x width
Area of the base = 20 cm x 18 cm
Area of the base = 360 cm squared
Then, calculate the area of each triangular face, which can be found using the formula:
1/2 x base x height
For the triangular face with a base length of 20 cm and height of 16.6 cm:
Area = 1/2 x 20 cm x 16.6 cm
Area = 166 cm squared
For the triangular face with a base width of 18 cm and height of 17.2 cm:
Area = 1/2 x 18 cm x 17.2 cm
Area = 155.4 cm squared
Add the areas of the base and four triangular faces together to find the total surface area of the rectangular pyramid:
Surface area = Base area + 4 x (triangle area)
Surface area = 360 cm squared + 4 (166 cm squared + 155.4 cm squared)
Surface area = 360 cm squared + 4(321.4 cm squared)
Surface area = 360 cm squared + 1,285.6 cm squared
Surface area = 1,645.6 cm squared
Therefore, the surface area of the rectangular pyramid is 1,645.6 cm squared, which is closest to the option provided: 1,643.2 cm squared.
try again you are wrong
My apologies for the error in my previous response. Let's recalculate the surface area of the rectangular pyramid.
Surface area of a rectangular pyramid = Base area + (1/2 x perimeter of base x slant height)
1. Calculate the area of the base:
Base area = length x width
Base area = 20 cm x 18 cm
Base area = 360 cm²
2. Calculate the slant height of each triangular face using the Pythagorean theorem:
For the face with a base length of 20 cm and height of 16.6 cm:
Slant height = √(16.6² + 10²) ≈ √(276.56 + 100) ≈ √376.56 ≈ 19.4 cm
For the face with a base width of 18 cm and height of 17.2 cm:
Slant height = √(17.2² + 9²) ≈ √(295.84 + 81) ≈ √376.84 ≈ 19.4 cm
3. Calculate the perimeter of the base:
Perimeter of base = 2(length + width)
Perimeter of base = 2(20 cm + 18 cm)
Perimeter of base = 2(38 cm)
Perimeter of base = 76 cm
4. Calculate the surface area:
Surface area = Base area + (1/2 x Perimeter of base x slant height)
Surface area = 360 cm² + (1/2 x 76 cm x 19.4 cm)
Surface area = 360 cm² + (38 cm x 19.4 cm)
Surface area = 360 cm² + 737.2 cm²
Surface area = 1,097.2 cm²
Therefore, the correct surface area of the rectangular pyramid is 1,097.2 cm², which is closest to the option provided: 1,178.8 cm².
which is the correct answer
The correct answer is:
1,178.8 cm²