Question

# A statue, $1.6m$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60°$ and from the same point the angle of elevation of the top of the pedestal is $45°$. Find the height of the pedestal.

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Solution

## Let $AB$ be the height of the statue.$D$ is the point on the ground from where the elevation is taken.To find the eight of pedestal $=BC=AC-AB$From the figure AB $=statue=1.6m$$BC=pedestal=?$In $∆ACD$$\frac{P}{B}=\frac{AC}{CD}=\mathrm{tan}60°$$1.6+\frac{BC}{CD}=\sqrt{3}$$\sqrt{3}CD=1.6+BC$$CD=1.6+\frac{BC}{\sqrt{3}}---------------\left\{1\right\}$In $∆BCD,$$\frac{BC}{CD}=\mathrm{tan}45°\phantom{\rule{0ex}{0ex}}\frac{BC}{CD}=1$$CD=BC$From (1) $\frac{1.6+BC}{\sqrt{3}}=\frac{BC}{1}$$\sqrt{3}BC=1.6+BC$$1.732BC–1BC=1.6$$0.732BC=1.6$$BC=\frac{1.6}{0.732}$$BC=\frac{16}{10}×\frac{100}{732}=\frac{1600}{732}$$BC=2.18m$Hence, the height of the pedestal is$2.18m$.

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