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The largest of these five squares has side length 27 units, and each additional square's side length is 2/3 as the one before. What is the total area of all five squares? 

 Mar 29, 2023
 #1
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The area of the largest square is 27^2 = 729.

 

The total area is then 729 + 729 (2/3) + 729 (2/3)^2 + 729 (2/3)^3 + 729 (2/3)^4 = 729 + 486 + 324 + 216 + 144 = 1899.

 Mar 29, 2023
 #2
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27^2 ==729 

 

[27 *2/3]^2==324

 

[sqrt(324) * 2/3]^2 ==144

 

[Sqrt(144) * 2/3]^2 ==64

 

[Sqrt(64) * 2/3]^2 ==28 + 4/9

 

729 + 324 + 144 + 64 + 28 + 4/9 ==1,289 + 4/9 - combined area of the 5 squares.

 Mar 29, 2023
 #3
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27^2   +   2/3 * 27 * 2/3 * 27      + .......

=    27^2   +  (2/3 * 27)^2      +   (2/3)^4 * 27^2     + (2/3)^6  27^2   + (2/3)^8 * 27^2    =

(2/3^0  + 2/3 ^2  + 2/3 ^4 + 2/3 ^6 )  * 27^2    =   1289  4/9  units^2

 

 

 

Double check:

27^2   + 18^2   + 12^2   + 8^2   + (5 1/3)2  = 1289 4/9     <===CHECK ! 

 Mar 30, 2023

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