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Rectangle \(ABCD\) is the base of pyramid \(PABCD\). If \(AB=8\), \(BC=4\), \(\overline{PA}\perp \overline{AB}\), \(\overline{PA}\perp \overline{AD}\), and \(PA=6\), then what is the volume of \(PABCD\)?

 Mar 27, 2019
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The volume of the pyramid is (1/3) (base area) height

 

The base area =    AB * BC  = 8 * 4  =  32

 

We can find the length of  the base diagonal AC  as

 

√ [ AB^2 + BC^2 ]=   √ [ 8^2 + 4^2 ] = √ [ 64 + 16 ] = √80  = 4√5

 

And  half of this distance is   2√5 =  √20

 

So...the height  =  √[ PA^2 - (√200^2 ] =  √ [ 6^2 -  20  ]  =  √ [ 36 - 20 ]  = √ 16  =  4

 

So....the volume of the  pyramid is

 

(1/3)(32)(4) =

 

(1/3) 128   =

 

128 / 3  units^3   =    42 + 2/3   units^3

 

 

cool cool cool

 Mar 27, 2019

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