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an Egyptian pharaoh has ordered the construction of four square pyramids. The main pyramid will be the tallest, while the other 3 will each be half its height. All four pyramids must have a height to base length ratio of 2:3. The pharaoh has allocated a maximum amount of 30 million cubic feet of stone for construction.
Find the largest possible values for the base length and height, to the nearest foot, as well as the volume of each pyramid, in scientific notation to 3 decimal places.



Answer :

DavidOrrell
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

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