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Bot GPT 3.5 To find the surface area of a rectangular pyramid, we need to find the area of each face and add them together.
The base of the pyramid is a rectangle with length l = 13 cm and width w = 11 cm. So, the area of the base is l * w = 13 cm * 11 cm = 143 cm^2.
The other four faces are triangles. The height of the pyramid (from the apex to the center of the base) is h = 8 cm. The base of each triangle is the same as the length or width of the base rectangle, so the base of each triangle is either 13 cm or 11 cm. The height of each triangle depends on the slant height of the pyramid, which we can find using the Pythagorean theorem.
The slant height of the pyramid, s, is the hypotenuse of a right triangle with legs h (height of the pyramid) and r (the distance from the apex to one of the corners of the base rectangle). To find r, we can use the Pythagorean theorem: r^2 + (w/2)^2 = h^2. Plugging in the known values, we have:
r^2 + (11/2)^2 = 8^2
r^2 + 121/4 = 64
r^2 = 64 - 121/4
r^2 = 256/4 - 121/4
r^2 = 135/4
r = sqrt(135/4)
r = sqrt(135)/2
Now we can find the height of the triangles using the Pythagorean theorem:
h_triangle = sqrt(s^2 - r^2)
Let's find the height of each triangle. We will consider one triangle with a base length of 13 cm (the other triangle will have a base length of 11 cm, but the height will be the same).
base = 13 cm
r = sqrt(135)/2
s = sqrt(h^2 + r^2) = sqrt(8^2 + (sqrt(135)/2)^2)
h_triangle = sqrt(s^2 - r^2) = sqrt((sqrt(64 + 135/4))^2 - (sqrt(135)/2)^2)
Plugging in the values, we get:
h_triangle = sqrt(256 + 135) - sqrt(135)/2
Now we can find the area of each triangle:
area_triangle = (1/2) * base * height_triangle = (1/2) * 13 cm * h_triangle cm
So, the surface area of the pyramid is:
143 cm^2 (area of the base) + 2 * area_triangle (area of the two triangles with a base length of 13 cm) + 2 * area_triangle (area of the two triangles with a base length of 11 cm)
Let's plug in the values and calculate:
surface area = 143 cm^2 + 2 * (1/2) * 13 cm * h_triangle cm + 2 * (1/2) * 11 cm * h_triangle cm
surface area ≈ 143 cm^2 + 13 cm * h_triangle cm + 11 cm * h_triangle cm
Now we just need to calculate the approximate value for h_triangle:
h_triangle = sqrt(256 + 135) - sqrt(135)/2 ≈ 11.6 cm
Plugging this value into the surface area formula, we have:
surface area ≈ 143 cm^2 + 13 cm * 11.6 cm + 11 cm * 11.6 cm
surface area ≈ 143 cm^2 + 150.8 cm^2 + 127.6 cm^2
surface area ≈ 421.4 cm^2
Therefore, the surface area of the rectangular pyramid is approximately 421.4 cm^2.