Q8. A statue 1.6 m tall stands on the top a pedestal. From a point on the ground the angle of elevation of the top of the statue is 60º and from the same point the angle of elevation of the top of the pedestal is 45º. Find the height of the pedestal.

Let BC be the pedestal of height h metres and CD be the statue of height 1.6 m. Let A be a point on the ground such that $$∠ \ CAB \ = \ 45º$$ and$$∠ \ DAB \ = \ 60º$$

In $$∆ \ ABC$$ and $$∆ \ ABD$$, we have

$$\frac{AB}{BC} \ = \ cot45º \ => \ \frac{AB}{h} \ = \ 1$$ $$=> \ AB \ = \ h$$      (1)

and, $$\frac{BD}{AB} \ = \ tan60º \ => \ \frac{BC + CD}{AB} \ = \ \sqrt{3}$$ $$=> \ \frac{h+1.6}{h} \ = \ \sqrt{3} \ => \ h+1.6 \ = \ h \sqrt{3}$$

=> $$h( \sqrt{3} \ - \ 1) \ = \ 1.6 \ => \ h \ = \ \frac{1.6}{ \sqrt{3} \ - \ 1} \ = \ \frac{1.6}{ \sqrt{3} \ - \ 1} × \frac{ \sqrt{3} \ + \ 1}{ \sqrt{3} \ + \ 1}$$

$$= \ \frac{1.6( \sqrt{3} \ + \ 1)}{2} \ = \ 0.8( \sqrt{3}+1)$$ $$= \ 2.1856$$

∴ the height of the pedestal is $$2.1856$$ m