What is the maximum volume of a pyramid that can fit inside a cube with a side length of 12 cm?
To find the maximum volume of a pyramid that can fit inside a cube, we need to consider the dimensions of the pyramid that would give the largest volume.
Let's assume that the pyramid is a square pyramid with a square base of side length x. Since the pyramid has to fit inside a cube with a side length of 12cm, we know that x ≤ 12.
The height of the pyramid, h, can be found using the Pythagorean theorem applied to the right triangle formed by the height, half the side of the base, and a side edge of the pyramid. We have:
h^2 + (x/2)^2 = x^2
h^2 = x^2 - (x/2)^2
h^2 = 3x^2/4
h = √(3x^2/4)
The volume of the pyramid is given by:
V = (1/3) * x^2 * h
Substituting h:
V = (1/3) * x^2 * √(3x^2/4)
Simplifying:
V = √3 x^3 / 12
To find the maximum volume, we need to find the value of x that gives the maximum value of V. Since x ≤ 12, we can take the derivative of V with respect to x and set it to zero:
dV/dx = (√3/4) * x^2 - (√3/8) * x^2 = (√3/8) * x^2
Setting dV/dx = 0:
(√3/8) * x^2 = 0
x = 0 or x ≠ 0
Since x has to be greater than 0, we have x ≠ 0. Therefore, the only critical point is at x = 0.
Now we need to check the endpoints of the interval [0, 12]. We have:
V(0) = 0
V(12) = √3 * 12^3 / 12 = 144√3
Therefore, the maximum volume of a square pyramid that can fit inside a cube with a side length of 12cm is 144√3 cubic centimeters.
To find the maximum volume of a pyramid that can fit inside a cube, we need to consider the dimensions and the shape of the pyramid.
In this case, the pyramid must fit inside a cube with a side length of 12 cm. The base of the pyramid will be the same as the base of the cube, which is a square with a side length of 12 cm.
Let's assume that the height of the pyramid is 'h'.
The volume of a pyramid can be calculated using the formula:
Volume = (1/3) * Base Area * Height
Since the base of the pyramid is a square, the base area is calculated by multiplying the length of one side of the base by itself:
Base Area = side length * side length
Given that the side length of the base is 12 cm, the base area is:
Base Area = 12 cm * 12 cm = 144 cm^2
Therefore, the volume of the pyramid is:
Volume = (1/3) * Base Area * Height = (1/3) * 144 cm^2 * h = 48 cm^2 * h
To find the maximum volume, we need to find the maximum value for 'h' that allows the pyramid to fit entirely inside the cube.
The maximum height 'h' is equal to the length of the space from the center of the cube to one of the corners.
Since the cube has a side length of 12 cm, the diagonal of the cube can be calculated using the Pythagorean theorem:
Diagonal = sqrt(12^2 + 12^2 + 12^2) = sqrt(432) ≈ 20.785 cm
Therefore, the maximum height 'h' can be calculated as half of the diagonal:
h = Diagonal / 2 = 20.785 cm / 2 = 10.3925 cm
Now, we can substitute the value of 'h' into the volume equation to find the maximum volume:
Volume = 48 cm^2 * h = 48 cm^2 * 10.3925 cm = 499.8298 cm^3
Hence, the maximum volume of a pyramid that can fit inside a cube with a side length of 12 cm is approximately 499.8298 cm^3.