Question

A statue 1.6 m tall stands on the top of 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.

Answer 1

The height of the pedestal is :

Given : Height of the statue, RS = 1.6 m, angle of elevation of top of the statue = 60°, angle of elevation of top of the pedestal = 45°

• Let suppose point is at a distance x m from foot of the pedestal and height of pedestal be h m.

• In ∆PQR,

tan 45° = h / x

1 = h / x

h = x

• In ∆PQS,

tan 60° = 1.6 + h / x

√3x = 1.6 + h

√3h = 1.6 + h

h(√3 - 1) = 1.6

• h = 1.6 / (√3-1)

= 1.6(√3-1) / 2 = 0.8(√3-1) m

Answer 2

$\large\sf\red{In\:∆BCD,}$

$\large\sf\red{tan\:45°=\frac{BC}{DC}}$

$\large\sf\red{1=\frac{h}{DC}}$

$\therefore$$\large\sf\red{DC=h}$

⠀⠀

$\large\sf\purple{In\:∆ACD,}$

$\large\sf\purple{\angle{ADC}=60°}$

$\large\sf\purple{tan\:60°=\frac{AC}{DC}}$

$\large\sf\purple{√3=\frac{1.6+h}{h}}$

$\large\sf\purple{√3h=1.6+h}$

$\large\sf\purple{√3h-h=1.6}$

$\large\sf\purple{h(√3-1)=1.6}$

$\large\sf\purple{h=\frac{1.6}{√3-1}}$

$\large\sf\purple{h=\frac{1.6}{√3-1}×\frac{√3+1}{√3+1}}$

$\large\sf\purple{h=\frac{1.6(√3+1)}{2}}$

$\large\sf\purple{h=0.8(√3+1)m}$