10. What is the maximum volume of a square pyramid that can fit inside a cube with a side length of 14 cm?
pls help!! I'm behind and I don't understand. I would just like to know how to solve it please?
the base of the pyramid is the face of the cube ... 14 cm on a side
the height of the pyramid is the side length of the cube ... also 14 cm
volume of pyramid = 1/3 * base area * height
thankyou so much!!!
To find the maximum volume of a square pyramid that can fit inside a cube, we need to understand a few concepts.
First, let's visualize a square pyramid inscribed in a cube. The base of the square pyramid is a square that is the same size as one face of the cube. The height of the pyramid is the perpendicular distance from the base to the apex of the pyramid.
To maximize the volume of the pyramid, we need to ensure that it fills the entire space inside the cube. In other words, all the edges of the pyramid need to touch the edges of the cube.
Now, let's break down the steps to find the maximum volume:
Step 1: Find the diagonal length of the base of the square pyramid.
Since the base of the pyramid is a square, we can use the Pythagorean theorem to find its diagonal length. The diagonal length (d) of a square can be calculated using the formula d = √(2s^2), where s is the side length of the square. In this case, the side length of the square is equal to the side length of the cube (14 cm).
d = √(2 * 14^2) = √(2 * 196) = √(392) ≈ 19.80 cm
Step 2: Calculate the height of the pyramid.
To maximize the volume, the height of the pyramid needs to be such that it touches the apex of the pyramid to the center of one of the square faces of the cube. In this case, the height is equal to half of the diagonal length of the base.
height = d/2 = 19.80 cm / 2 = 9.90 cm
Step 3: Calculate the volume of the pyramid using the formula:
Volume = (1/3) * (side length of base)^2 * height
Volume = (1/3) * (14 cm)^2 * 9.90 cm
Volume = (1/3) * 196 cm^2 * 9.90 cm
Volume ≈ 650.80 cm^3
Therefore, the maximum volume of the square pyramid that can fit inside the given cube is approximately 650.80 cubic centimeters.
You would want the base of the square pyramid to fit exactly on
the base of the cube, and since the volume of the pyramid now
depends only on its height, make the height 14.
Max volume of pyramid = (1/3)(base)(height)
= (1/3)(14)^2 (14)
= (1/3)(14^3) = 2744/3 cm^3
btw, it really doesn't matter where the vertex of the pyramid is, as
long as it is 14 cm above the base, the volume would always be the same.