Question

A statue which is x m tall stands on the top of 100m long pedestal on the ground. From

a point on the ground, the angle of elevation of the top of the statue is 600

and from the

same point, the angle of elevation of the top of the pedestal is 450

. Find the height of

the statue.​

Answer 1

$\begin{gathered}\Large\bf{\color{indigo}GiVeN,} \\ \end{gathered}$

Length of a statue is x m

Statue stands on the top of 100 m long pedestal.

A point on the ground, the angle of elevation of the top of the statue is 60°

From the same point, the angle of elevation of the top of the pedestal is 45°.

$\begin{gathered}\Large\bf{\color{cyan}To\:Find,} \\ \end{gathered}$

The height of the statue.

$\begin{gathered}\Large\bf{\color{lime}CaLcUlAtIoN,} \\ \end{gathered}$

☆ See the attachment diagram.

$\begin{gathered}\bf\red{As\:shown\:in\:diagram,} \\ \end{gathered}$

AB is a Pedestal.

AC is a Statue.

$\begin{gathered}\bf\pink{Given\:that,} \\ \end{gathered}$

Height of the Pedestal (AB) = 100m

$\begin{gathered}\bf\blue{According\:to\:the\:question,} \\ \end{gathered}$

Height of the Statue (AC) = 'x'

Angle of elevation to top of statue = 60°

$\begin{gathered}\longmapsto\:\:\bf{\angle{CPB}\:=\:60°} \\ \end{gathered}$

Angle of elevation to top of pedestal = 45°

$\begin{gathered}\longmapsto\:\:\bf{\angle{APB}\:=\:45°} \\ \end{gathered}$

✅ Since statue is perpendicular to the ground,

$\begin{gathered}\longmapsto\:\:\bf{\angle{ABP}\:=\:90°} \\ \end{gathered}$

$\begin{gathered}\bf\blue{In\:ABP\:right\:angle\:triangle,} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{tan{45}\:=\:\dfrac{AB}{BP}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{1\:=\:\dfrac{AB}{BP}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf\orange{AB\:=\:BP\:=\:100\:m} \\ \end{gathered}$

$\begin{gathered}\bf\pink{In\:PCB\:right\:angle\:triangle,} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{tan{60}\:=\:\dfrac{BC}{BP}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{\sqrt{3}\:=\:\dfrac{BC}{100}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{BC\:=\:\sqrt{3}\times{100}\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{BC\:=\:100{\sqrt{3}}\:m\:} \\ \end{gathered}$

$\begin{gathered}\bf\red{We\:have,} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{Height\:of\:statue\:(AC)\:=\:BC\:-\:AB\:} \\ \end{gathered}$

$\begin{gathered}:\implies\:\:\bf{Height\:of\:statue\:(AC)\:=\:100{\sqrt{3}}\:-\:100\:} \\ \end{gathered}$

$\begin{gathered}:\implies\::</p><p>\bf\green{Height\:of\:statue\:(AC)\:=\:100({\sqrt{3}}\:-\:1)\:m\:} \\ \end{gathered}$

$\Large\bold\therefore∴ The height of the statue is \bf{\color{coral}100({\sqrt{3}}\:-\:1)\:m}$