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(C) Let the height of the house be $h \ m$. The distance from the foot of the house is $d = 70 \ m$. The height of the flag-post is $20 \ m$.Let $	he

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$50$

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(C) Let the height of the house be $h \ m$. The distance from the foot of the house is $d = 70 \ m$. The height of the flag-post is $20 \ m$.Let $	heta$ be the angle subtended by the house at the point,and $\phi$ be the angle subtended by the house plus the flag-post.Then $	an 	heta = \frac{h}{70}$ and $	an \phi = \frac{h + 20}{70}$.The angle subtended by the flag-post is $\alpha = \phi - 	heta$,where $	an \alpha = \frac{1}{6}$.Using the formula $	an \alpha = \frac{	an \phi - 	an 	heta}{1 + 	an \phi 	an 	heta}$:$\frac{1}{6} = \frac{\frac{h + 20}{70} - \frac{h}{70}}{1 + (\frac{h + 20}{70})(\frac{h}{70})}$$\frac{1}{6} = \frac{\frac{20}{70}}{1 + \frac{h(h + 20)}{4900}}$$1 + \frac{h^2 + 20h}{4900} = 6 	imes \frac{20}{70} = \frac{120}{70} = \frac{12}{7}$$\frac{h^2 + 20h}{4900} = \frac{12}{7} - 1 = \frac{5}{7}$$h^2 + 20h = \frac{5}{7} 	imes 4900 = 5 	imes 700 = 3500$$h^2 + 20h - 3500 = 0$$(h + 70)(h - 50) = 0$Since $h > 0$,we have $h = 50 \ m$.

$A$ flag-post $20 \ m$ high standing on the top of a house subtends an angle whose tangent is $\frac{1}{6}$ at a distance $70 \ m$ from the foot of the house. The height of the house is......$m$

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