The pair of square pyramids are similar.
v=64in^3
v=343in^3
To find the ratio of the volumes of the two similar square pyramids, we will take the square root of both volumes to find the side length of each pyramid.
For the first pyramid with volume 64in^3:
v = (1/3)(s^2)(h)
64 = (1/3)(s^2)(h)
Since this pyramid is a square pyramid, the base is a square. Let the side length of the base be x. The height can be found using the Pythagorean theorem which gives
h = (x^2 - (x/2)^2)^(1/2)
h = (3/4)x
Substitute h into the volume equation
64 = (1/3)(x^2)((3/4)x)
192 = x^3
x = 6 in
For the second pyramid with volume 343in^3:
v = (1/3)(s^2)(h)
343 = (1/3)(s^2)(h)
Finding the side length of the base as y:
343 = (1/3)(y^2)(3y)
343 = y^3
y = 7 in
The ratio of the volumes of the two similar square pyramids is:
64/343 = 4/7
Therefore, the ratio of their volumes is 4:7.