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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

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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?

Let h = height of the pyramid

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

$$\mathbf{h=\ ? }$$

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

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

Jan 29, 2019