Let's call the height of the tallest pyramid x:
- The base length is 3/2 * x (following the ratio) = 3x/2
- This makes the volume, which follows the general formula V = 1/3 * base area * height = 1/3 * (3x/2)^2 * x = 1/3 * (9x^2)/4 * x = (3x^3)/4
- The height of the smaller pyramids is x/2
- The base length of these will be 3/2 * x/2 = 3x/4
- This makes the volume, which also follows the formula above, 1/3 * (3x/4)^2 * x/2 = 1/3 * (9x^2)/16 * x/2 = (3x^3)/32
- There are three smaller pyramids and one larger one, so the total volume is:
(3x^3)/4 + 3(3x^3)/32 = 30,000,000
This can be simplified by getting both algebraic fractions over the same denominator (namely getting the first over 32 instead of 4, by multiplying it by 8):
8(3x^3)/32 + 3(3x^3)/32 = 30,000,000
11(3x^3)/32 = 30,000,000
11(3x^3 ) = 30,000,000 * 32 = 960,000,000
33x^3 = 960,000,000
x^3 = 960,000,000 / 33 = 29090909.0909
x = cubic root of 29090909.0909 = 307.5523839ft
Now you just need to work out the dimensions and volumes of each pyramid:
LARGE
Height = x = 307.5523839ft = 3.075523839 * 10^2 ≈ 3.076 * 10^2 ft
Base length = 3x/2 = (3 * 307.5523839)/2 = 461.3285759 ≈ 4.613 * 10^2 ft
Volume = (3x^3)/4 = (3 * 307.5523839^3)/4 = 21818181.81 ≈ 2.182 * 10^7 ft^3
SMALL
Height = x/2 = 307.5523839/2 = 153.776192 ≈ 1.538 * 10^2 ft
Base length = 3x/4 = (3 * 307.5523839)/4 = 230.6642879 ≈ 2.307 * 10^2 ft
Volume = (3x^3)/32 = (3 * 307.5523839^3)/32 = 2727272.726 ≈ 2.727 * 10^6 ft^3
CHECK
(2.182 * `10^7) + 3(2.727 * 10^6) = 30,001,000 ≈ 30,000,000 ft^3 (rounding error)
I hope this helps
