Answer:
a) Height of the pyramid is 3 cm.
b) Height of 12 cm and length of 16 cm
Explanation:
A) Finding the height of the pyramid with square base and given volume
The volume V of a pyramid is given by the formula:
V = 1/3 * (base area) * height
The base area A is a square with side length of 4 cm, so:
A = side_length^2 = 4^2 = 16 cm^2
So, we can solve the volume formula for height:
height = V / (1/3 * A) = 16 cm^3 / (1/3 * 16 cm^2)
Doing the calculation, we find:
height = 16 cm^3 / (5.333 cm^2) = 3 cm
So, the height of the pyramid is 3 cm.
B) Finding the dimensions of a similar pyramid with a given volume
The volume of a similar pyramid is proportional to the cube of the scale factor (because volume is a three-dimensional measurement). So, if we let k be the scale factor, then:
V2 = k^3 * V1
We can solve for k:
k = (V2 / V1)^(1/3)
Then we can find the height and side length of the base of the similar pyramid as:
height2 = k * height1
side_length2 = k * side_length1
In this case, V1 is 16 cm^3, V2 is 1024 cm^3, height1 is 3 cm, and side_length1 is 4 cm. So:
k = (1024 cm^3 / 16 cm^3)^(1/3) = 4
So:
height2 = 4 * 3 cm = 12 cm
side_length2 = 4 * 4 cm = 16 cm
So, the similar pyramid with a volume of 1024 cm^3 has a height of 12 cm and a base side length of 16 cm.