A 7 m long flagstaff is fixed on the top of a tower. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 45° and 36° respectively. Find the height of the tower correct to one place of decimal.

Let CD be the tower of height h meters and BD the flagstaff.

A be point on the ground from where the angles of elevation of the top and bottom of the flagstaff are 45° and 36° respectively.

From figure,

BC = BD + DC = (7 + h) meters.

Considering right angled triangle △ABC,

$\Rightarrow \text{tan 45°} = \dfrac{BC}{AC} \\[1em] \Rightarrow 1 = \dfrac{7 + h}{AC} \\[1em] \Rightarrow AC = h + 7 \text{ ……(Eq 1)}.$

Considering right angled triangle △ADC,

$\Rightarrow \text{tan 36°} = \dfrac{DC}{AC} \\[1em] \Rightarrow 0.7265 = \dfrac{h}{AC} \\[1em] \Rightarrow AC = \dfrac{h}{0.7265}$

Putting value of AC in Eq 1 we get,

$\Rightarrow \dfrac{h}{0.7265} = h + 7 \\[1em] \Rightarrow h = 0.7265(h + 7) \\[1em] \Rightarrow h = 0.7265h + 5.0855 \\[1em] \Rightarrow h - 0.7265\text{ h} = 5.0855 \\[1em] \Rightarrow 0.2735\text{ h} = 5.0855 \\[1em] \Rightarrow h = \dfrac{5.0855}{0.2735} \\[1em] \Rightarrow h = 18.6 \text{ m}.$

Hence, the height of tower is 18.6 m.