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Heights and Distances — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsHeights and Distances3 Marks
Question
A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff. At a point on the plane $70$ metres away from the tower, an observer notices that the angles of elevation of the top and the bottom of the flagstaff are respectively $60^\circ $ and $45^\circ $. Find the height of the flag-staff and that of the tower.
✓

Answer

Let $BC$ be the tower of height  $x\ m$ and $AB$ be the flag staff of height $y, 70\ m$ away from the tower, makes an angle of elevation are $60^\circ$ and $45^\circ $ respectively from top and bottom of the flag staff.
Let $AB = y\ m, BC = x\ m$ and $CD = 70\ m.$
$\angle\text{ADC}=45^\circ$ and $\angle\text{ADC}=60^\circ$
So we use trigonometric ratios.
In a triangle $BCD,$
$\Rightarrow\ \tan\text{D}=\frac{\text{BC}}{\text{CD}}$
$\Rightarrow\ \tan45^\circ=\frac{\text{x}}{70}$
$\Rightarrow\ 1=\frac{70}{\text{x}}$
$\Rightarrow\ \text{x}=70$
Again in a triangle $ADC,$
$\Rightarrow\ \tan\text{D}=\frac{\text{AB}+\text{BC}}{\text{CD}}$
$\Rightarrow\ \sqrt{3}=\frac{\text{y}+70}{70}$
$\Rightarrow\ 70\sqrt{3}=70+\text{y}$
$\Rightarrow\ \text{y}=70(\sqrt{3}-1)$
$\Rightarrow\ \text{y}=51.24$
Hence the height of flag staff is $51.24\ m$ and height of tower is $70\ m.$

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