Two pyramids are similar.
The surface area of the first is 250 and the surface area of the second is 22.5.
The heights of the pyramids are in a ratio of _ : _
The volumes of the pyramids are in a ratio of _ : _
Two pyramids are similar.
area:area = (height:height)^2
volume:volume = (height:height)^3
If the scale factor (ratio between linear
measurements of two similar figures) is k then:
The ratio between area of those two similar figures will be k²
The ratio between volume of those two similar figures will be k³
The ratio between area = 250 / 22.5 = k²
k = √ ( 250 / 22.5 ) = √ ( 2.5 ∙ 100 / 2.5 ∙ 9 ) =
√ ( 100 / 9 ) = √ 100 / √ 9 = 10 / 3
The heights of the pyramids is linear measurements.
The heights of the pyramids are in a ratio of k = 10 : 3
The volumes of the pyramids are in a ratio of k³ = 10³ : 3³ = 1000 : 27
To find the ratio of the heights of the two similar pyramids, we can use the fact that the surface area of a pyramid is proportional to the square of its height.
Let's say the heights of the first and second pyramids are h1 and h2, respectively. We can set up the following proportion:
(surface area of first pyramid) / (surface area of second pyramid) = (height of first pyramid)^2 / (height of second pyramid)^2
Plugging in the given values, we have:
250 / 22.5 = h1^2 / h2^2
Simplifying the equation:
(250 * h2^2) / 22.5 = h1^2
Now, to find the ratio of the heights, we need to take the square root of both sides of the equation:
sqrt[(250 * h2^2) / 22.5] = sqrt[h1^2]
Simplifying further:
(h2 / sqrt(22.5)) * sqrt(250) = h1
Therefore, the ratio of the heights of the two pyramids is:
h1 : h2 = (h2 / sqrt(22.5)) * sqrt(250) : h2
To find the ratio of the volumes of the two similar pyramids, we can use the fact that the volume of a pyramid is proportional to the cube of its height.
Let's say the volumes of the first and second pyramids are V1 and V2, respectively. We can set up the following proportion:
(volume of first pyramid) / (volume of second pyramid) = (height of first pyramid)^3 / (height of second pyramid)^3
Plugging in the given values, we have:
V1 / V2 = h1^3 / h2^3
Therefore, the ratio of the volumes of the two pyramids is:
V1 : V2 = h1^3 : h2^3