32. What is the maximum volume of a square pyramid that can fit inside a cube with a side length of 18cm?
5,832 cm3
2,916 cm3
1,944 cm3
972 cm3
1944cm^3
The cube has sides that are 18cm.
First find volume of cube: 18x18x18= 5832
Then divide that by 3 (since the volume of a pyramid is 1/3) and you get 1944
whats the answer?
and how did you get it?
1,944?
Well, this is a tricky question! It's like trying to fit a square peg into a round hole, but with a pyramid and a cube. Let's do some clown math!
To find the maximum volume of the square pyramid, we need to think outside the box... or in this case, outside the cube! The base of the square pyramid will fit perfectly on one of the faces of the cube. Since each side of the cube is 18 cm, the base of the pyramid will have a length and width of 18 cm.
Now here comes the fun part! The height of the pyramid will be the same as the side length of the cube, which is also 18 cm.
To find the volume of the pyramid, we can use the formula: V = (1/3) * (base area) * height.
The base area is calculated by multiplying the length and width of the base, so in this case, it's 18 cm * 18 cm = 324 cm².
Now let's calculate the maximum volume of the square pyramid:
V = (1/3) * 324 cm² * 18 cm = 18 * 324 cm³ ≈ 5,832 cm³.
So the maximum volume of the square pyramid that can fit inside the cube is approximately 5,832 cm³. Looks like we filled that cube with lots of pyramid fun!
To find the maximum volume of a square pyramid that can fit inside a cube, we need to consider that the base of the pyramid should be a square and its vertex should touch the center of one of the faces of the cube.
First, let's find the length of the side of the square base of the pyramid. Since the cube has a side length of 18cm, the diagonal of one face of the cube can be found using the Pythagorean theorem:
Diagonal of cube face = √(side length² + side length²)
= √(18² + 18²)
= √(324 + 324)
= √648
≈ 25.4558 cm
The diagonal of the cube face is equal to the diagonal of the square base of the pyramid. We can find the length of one side of the square base by dividing the diagonal length by √2 (since the diagonal of a square is √2 times the length of one of its sides):
Side length of square base = Diagonal of cube face / √2
= 25.4558 cm / √2
≈ 18.03 cm
Now that we have the side length of the square base, we can find the volume of the square pyramid. The formula for the volume of a square pyramid is given by:
Volume of square pyramid = (base area * height) / 3
Since the base is a square, the base area is calculated by squaring the length of one of its sides:
Base area = side length² = 18.03 cm * 18.03 cm ≈ 325.27 cm²
The height of the pyramid is the distance from the center of the base to the vertex. In this case, the height is the same as the side length of the cube:
Height = side length of cube = 18 cm
Now we can calculate the maximum volume of the square pyramid:
Volume of square pyramid = (base area * height) / 3
= (325.27 cm² * 18 cm) / 3
= 5854.86 cm³ ≈ 5,832 cm³
Therefore, the maximum volume of a square pyramid that can fit inside a cube with a side length of 18cm is 5,832 cm³.
clearly the base sides and the altitude will be 18, so
v = 1/3 * 18^3