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(D) Let the height of the tower be $H$ and the foot of the tower be $P$.From point $A$,which is due east of the tower,the angle of elevation is $45^\c

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$54$

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(D) Let the height of the tower be $H$ and the foot of the tower be $P$.From point $A$,which is due east of the tower,the angle of elevation is $45^\circ$. Thus,$AP = H \cot 45^\circ = H$.From point $B$,which is due south of $A$,the angle of elevation is $30^\circ$. Thus,$BP = H \cot 30^\circ = H\sqrt{3}$.Since $A$ is due east and $B$ is due south of $A$,$	riangle PAB$ is a right-angled triangle at $A$ (where $PA \perp AB$).Using the Pythagorean theorem in $	riangle PAB$:$AB^2 + AP^2 = BP^2$$(54\sqrt{2})^2 + H^2 = (H\sqrt{3})^2$$5832 + H^2 = 3H^2$$2H^2 = 5832$$H^2 = 2916$$H = \sqrt{2916} = 54 \, 	ext{m}$.

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