the volume of 2 similar pyramid are in the ratio of 216:27 . what is the ratio of their a)height, b) diagonal?

volume of similar shapes is proportional to the cubes of their sides

so we need
ratio of their sides = 216^(1/3) : 27^(1/3)
= 6: 3
= 2 : 1

we could have seen that right away, since 216 : 27 = 8 : 1

since both height and diagonal are linear measurements, both of
these have that ratio
that is:
height of larger : height of smaller = 2 : 1
etc

Oh, it's pyramid time! Let's see what we've got here.

a) The ratio of the volumes is 216:27. To find the ratio of their heights, we need to take the cube root of that volume ratio. So, the cube root of 216 is 6, and the cube root of 27 is 3. Therefore, the ratio of their heights is 6:3, which simplifies to 2:1. It's like a short and tall pyramid hanging out together!

b) Now, let's find the ratio of their diagonals. The diagonal of a pyramid is a bit tricky, but we can do it! Since the pyramids are similar, their ratios of side lengths are the same as their ratios of heights. So, the ratio of their side lengths would be the same as the ratio of their heights, which is 2:1. Now, the diagonal is formed by connecting the apex to the midpoint of the base, so it's kind of like stretching out the height. Therefore, the ratio of their diagonals would also be 2:1.

So, to sum it up, the ratio of their heights is 2:1 (short and tall), and the ratio of their diagonals is also 2:1 (stretching it out). May your pyramids always stand tall and diagonals be perfectly ratioed!

To find the ratio of the height and diagonal of two similar pyramids, we can use the property that the ratio of volumes of two similar solids is equal to the cube of the ratio of their corresponding lengths.

Let's denote the ratio of the volumes of the two pyramids as V₁/V₂ = 216/27. Simplifying, we have V₁/V₂ = 8.

a) Ratio of heights:
Since the ratio of volumes is equal to the cube of the ratio of their corresponding lengths, we have (h₁/h₂)³ = V₁/V₂.
Substituting the given values, we get (h₁/h₂)³ = 8.
Taking the cube root on both sides, we find h₁/h₂ = 2.

Therefore, the ratio of their heights is 2:1.

b) Ratio of diagonals:
The ratio of volumes is equal to the cube of the ratio of their corresponding lengths. Similarly, the ratio of surface areas of two similar solids is equal to the square of the ratio of their corresponding lengths.

Let's denote the ratio of the diagonals of the two pyramids as d₁/d₂.
Since the surface area ratio is equal to the square of the ratio of their corresponding lengths, we have (d₁/d₂)² = V₁/V₂.
Substituting the given values, we get (d₁/d₂)² = 8.
Taking the square root on both sides, we find d₁/d₂ = √8 = 2√2.

Therefore, the ratio of their diagonals is 2√2:1.

To find the ratio of the height and diagonal of two similar pyramids, we need to know that the volume of similar solids is proportional to the cube of their dimensions.

Let's denote the volume of the first pyramid as V₁ and the volume of the second pyramid as V₂. The given ratio of their volumes is 216:27. We can set up the proportion:

V₁/V₂ = 216/27

Simplifying, we get:

V₁/V₂ = 8/1

This means that the volume of the first pyramid is 8 times the volume of the second pyramid.

a) Ratio of the height:
Since the volume is proportional to the cube of the height, we can find the ratio of their heights by taking the cube root of the volume ratio.
Let's denote the height of the first pyramid as h₁ and the height of the second pyramid as h₂.

(h₁/h₂)³ = 8/1

Taking the cube root of both sides, we get:

h₁/h₂ = ∛(8/1)

Simplifying, we find that the ratio of their heights is:

h₁/h₂ = 2/1

Therefore, the ratio of their heights is 2:1.

b) Ratio of the diagonals:
Since the volume is proportional to the cube of the diagonal, we can find the ratio of their diagonals by taking the cube root of the volume ratio.
Let's denote the diagonal of the first pyramid as d₁ and the diagonal of the second pyramid as d₂.

(d₁/d₂)³ = 8/1

Taking the cube root of both sides, we get:

d₁/d₂ = ∛(8/1)

Simplifying, we find that the ratio of their diagonals is:

d₁/d₂ = 2/1

Therefore, the ratio of their diagonals is also 2:1.