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Rectangle ABCD is the base of pyramid PABCD. If AB = 8, BC = 4, \(\overline{PA}\perp \overline{AD}\), \(\overline{PA}\perp \overline{AB}, \) and PB = 17, then what is the volume of PABCD?

 Jan 29, 2019
edited by Lightning  Jan 29, 2019
 #1
avatar+21191 
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NVM. Answer is 160.

Rectangle ABCD is the base of pyramid PABCD.

If AB = 8, BC = 4,\(\overline{PA}\perp \overline{AD}\), \( \overline{PA}\perp \overline{AB}\),  and PB = 17,

then what is the volume of PABCD?

 

Let h = height of the pyramid

Let B = Area of the Base of the pyramid = Area of the rectangle ABCD = \(4\cdot 8\) = \(32\)

 

\(\mathbf{h=\ ? }\)

\(\begin{array}{|rcll|} \hline h^2 + AB^2 &=& PB^2 \\ h^2 + 8^2 &=& 17^2 \\ h^2 &=& 17^2-8^2 \\ h^2 &=& 289-64 \\ h^2 &=& 225 \\ \mathbf{h} & \mathbf{=}& \mathbf{15} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \text{Volume of PABCD} &=& \dfrac{1}{3}\cdot B \cdot h \\\\ &=& \dfrac{1}{3}\cdot 32 \cdot 15 \\\\ &=& 5\cdot 32 \\\\ \mathbf{\text{Volume of PABCD}} & \mathbf{=}& \mathbf{160} \\ \hline \end{array}\)

 

laugh

 Jan 29, 2019

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