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What is the missing length of rectangular prism if height and width are both 8 inches and surface area is 256 in ^2 show your work
To find the missing length of the rectangular prism, we first need to determine the surface area of the rectangular prism using the formula:
Surface Area = 2lw + 2lh + 2wh
Given:
Height (h) = 8 inches
Width (w) = 8 inches
Surface Area = 256 square inches
Let the missing length be represented by 'L'.
Surface Area = 2(8)(L) + 2(8)(8) + 2(8)(8)
256 = 16L + 128 + 128
256 = 16L + 256
16L = 256 - 256
16L = 0
L = 0 / 16
L = 0
Therefore, the missing length of the rectangular prism is 0 inches.
Surface area of a square pyramid is 160cm^2. The area of the lateral triangles is 96cm^2. What is the length of one of the sides?
Show your work
Let's denote the side length of the square base of the pyramid as 's', and the slant height of the pyramid as 'l'.
Given:
Surface Area of the square pyramid (including the base) = 160 cm^2
Area of the lateral triangles = 96 cm^2
The total surface area of the pyramid can be expressed as the sum of the lateral surface area (4 triangles) and the base area:
Total Surface Area = Area of base + Area of lateral triangles
Given the lateral area of the triangles is 96 cm^2 and there are 4 triangles (each triangular face is an isosceles right triangle with base s and height l):
4(1/2 * base * height) = 96
2s √(s^2/4 + l^2) = 96
s √(s^2/4 + l^2) = 48
Given the total surface area of the pyramid is 160 cm^2:
160 = s^2 + 4(1/2 * s * l)
160 = s^2 + 2sl
Now we need to solve for 's' and 'l'.
From the first equation (s √(s^2/4 + l^2) = 48), we will find the value of 'l', and then substitute it into the second equation to solve for 's'.
From the second equation (160 = s^2 + 2sl), rewrite it in terms of 'l':
160 = s^2 + 2s(48/s)
160 = s^2 + 96
s^2 = 160 - 96
s^2 = 64
s = √64
s = 8 cm
Therefore, the length of one side of the square base of the pyramid is 8 cm.