How many times larger is a cube than a pyramid with the same base?
To find out how many times larger a cube is compared to a pyramid with the same base, we need to determine the volume of both shapes.
A cube has all sides equal in length, so if we denote the length of one side as "s", the volume of the cube can be calculated using the formula V_cube = s^3.
On the other hand, a pyramid has a base and a height, and the volume of a pyramid can be calculated using the formula V_pyramid = (1/3) * base_area * height.
Since the problem states that the cube and the pyramid have the same base, we can represent the base area as "B" and assume the height of the pyramid to also be "s" (since the sides of the pyramid are slanted).
Now, let's compare the two volumes:
V_cube = s^3
V_pyramid = (1/3) * B * s
To find out how many times larger a cube is than a pyramid, we divide the volume of the cube by the volume of the pyramid:
V_cube / V_pyramid = (s^3) / ((1/3) * B * s)
Simplifying the equation further, we can cancel out "s" from the numerator and denominator:
V_cube / V_pyramid = s^2 / (1/3 * B)
Finally, we can multiply the equation by 3 to get rid of the fraction:
3 * (V_cube / V_pyramid) = 3 * (s^2 / (1/3 * B))
3 * (V_cube / V_pyramid) = 9 * (s^2 / B)
Therefore, a cube is 9 times larger than a pyramid with the same base.