A cube of surface area X is sliced into two rectangular prisms.
One of the prisms has surface area 1/2X. What is the surface area
of the other prism?
each side of the cube has area x/6, so each edge has length √(x/6)
one prism, with height h has area 2(x/6) + 4h√(x/6)
So, we know that
x/3 + 4h√(x/6) = x/2
h = √(x/6)/4
The other prism, with height √(x/6)-h has area
x/3 + 4(√(x/6)-h)√(x/6)
= x/3 + 4(√(x/6)-√(x/6)/4)√(x/6)
= x/3 + 4(3/4 √(x/6))√(x/6)
= x/3 + 3(√(x/6))^2
= x/3 + 3(x/6)
= x/3 + x/2
= 5x/6
or, a little less brute-force, ...
Slicing the cube in two adds two more square faces
Each of the original faces has area x/6
Now we have a total area of
x + x/3 = 4x/3
one prism has area x/2, so the other has area
4x/3 - x/2 = 5x/6
nice
To find the surface area of the other prism, let's start by calculating the surface area of the entire cube.
Since a cube has equal-sized square faces, we can determine the surface area by multiplying the length of one side by itself and then multiplying that by 6 (since there are 6 faces in a cube).
Let's assume the length of one side of the cube is "s". So, the surface area of the entire cube is given by:
Surface Area = 6s^2 = X
Now, we need to find the surface area of each rectangular prism after slicing the cube.
One of the prisms has a surface area that is 1/2X. Let's denote the surface area of this prism as "A". So, we have:
A = 1/2X
Since the surface area of a rectangular prism is given by 2(lw + lh + wh), where l, w, and h are the length, width, and height of the prism, respectively, we can express the surface area of each prism as:
A = 2(lw + lh + wh)
Now, we can find the surface area of the other prism, denoted as "B". We know that the surface area of the entire cube (X) is equal to the sum of the surface areas of the two prisms (A + B), which can be written as:
X = A + B
Since we know the value of A (1/2X), we can substitute it into the equation:
X = 1/2X + B
To solve for B, we can subtract 1/2X from both sides:
X - 1/2X = B
Simplifying:
1/2X = B
Therefore, the surface area of the other prism (B) is 1/2X.