# Height and Distance Questions

FACTS  AND  FORMULAE  FOR  HEIGHT  AND  DISTANCE  PROBLEMS

1.In a right angled$\inline \fn_cm \Delta OAB$,where$\inline \fn_cm \angle BOA=\theta$

$\inline \fn_cm (i)\; sin\theta =\frac{Perpendicular}{Hypotenuse}=\frac{AB}{OB}$

$\inline \fn_cm (ii)\; cos\theta =\frac{Base}{Hypotenuse}=\frac{OA}{OB}$

$\inline \fn_cm (iii)\; tan\theta =\frac{Perpendicular}{Base}=\frac{AB}{OA}$

$\inline \fn_cm (iv)\; cosec\theta =\frac{1}{sin\theta }=\frac{OB}{AB}$

$\inline \fn_cm (v)\; sec\theta =\frac{1}{cos\theta }=\frac{OB}{OA}$

$\inline \fn_cm (vi)\; cot\theta =\frac{1}{tan\theta }=\frac{OA}{AB}$

2. Trigonometrical Identities :

$\inline \fn_cm (i)\; sin^2\theta +cos^2\theta =1$

$\inline \fn_cm (ii)\; 1+tan^2\theta =sec^2\theta$

$\inline \fn_cm (iii)\; 1+cot^2\theta =cosec^2\theta$

3. Values of T-ratios :

$\inline \fn_cm \begin{matrix} \theta & 0^{o}& 30^{o} & 45^{o} & 60^{o} &90^{o} \\ sin\theta & 0 & \frac{1}{2}& \frac{1}{\sqrt{2}} &\frac{\sqrt{3}}{2} & 1\\ cos\theta & 1 & \frac{\sqrt{3}}{2} &\frac{1}{\sqrt{2}} &\frac{1}{2} &0 \\ tan\theta & 0 &\frac{1}{\sqrt{3}} & 1 &\sqrt{3} & not\: defined \end{matrix}$

4. Angle of Elevation:

Suppose a man from a point O looks up at an object P, placed above the level of his eye. Then, the angle which the line of sight makes with the horizontal through O, is called the angle of elevation of P as seen from O.

$\inline \fn_cm \therefore$ Angle of elevation of P from O is =$\inline \fn_cm \angle AOP$

5. Angle of Depression :

Suppose a man from a point O looks down at an object P, placed below the level of his eye, then the angle which the line of sight makes with the horizontal through O, is called the angle of depression of P as seen from O.

Q:

If an object travels at five feet per second, how many feet does it travel in one hour?

 A) 30 B) 3000 C) 18 D) 1800

Explanation:

If an object travels at 5 feet per second it covers 5x60 feet in one minute, and 5x60x60 feet in one hour.

$\inline \fn_cm \therefore$Answer = 1800

13 3733
Q:

An observer 1.6 m tall is $\inline \fn_cm 20\sqrt{3}$ away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The heights of the tower is:

 A) 21.6 m B) 23.2 m C) 24.72 m D) None of these

Explanation:

Draw BE ${\color{Black}&space;\bot}$ CD

Then, CE = AB = 1.6 m,

BE = AC = ${\color{Black}&space;20\sqrt{3}m}$

${\color{Black}\frac{DE}{BE}=tan30^{0}=\frac{1}{\sqrt{3}}}$

${\color{Black}\Rightarrow&space;DE=\frac{20\sqrt{3}}{\sqrt{3}}m=20m}$

${\color{Black}\therefore&space;}$ CD = CE + DE = (1.6 + 20) m = 21.6 m.

10 3709
Q:

A man standing at a point P is watching the top of a tower, which makes an angle of elevation of 30º with the man's eye. The man walks some distance towards the tower to watch its top and the angle of the elevation becomes 60º. What is the distance between the base of the tower and the point P?

 A) Data inadequate B) 8 units C) 12 units D) None of these

Explanation:

One of AB, AD and CD must have given.

8 2814
Q:

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30º and 45º respectively. If the lighthouse is 100 m high, the distance between the two ships is:

 A) 173 m B) 200 m C) 273 m D) 300 m

Explanation:

Let AB be the lighthouse and C and D be the positions of the ships.

${\color{Black}&space;Then,AB=100m,&space;\angle&space;ACB=30^{\circ}&space;and&space;\angle&space;ADB=45^{\circ}}$

${\color{Black}&space;\frac{AB}{AC}=\tan&space;30^{\circ}=\frac{1}{\sqrt{3}}\Rightarrow&space;AC=AB\times&space;\sqrt{3}=100\sqrt{3}m}$

${\color{Black}&space;\frac{AB}{AD}=\tan&space;45^{\circ}=1\Rightarrow&space;AD=AB=100&space;m}$

${\color{Black}&space;\therefore&space;CD=(AC+AD)=(100\sqrt{3}+100)m=100(\sqrt{3}+1)=100\times&space;2.73=273&space;m}$

6 1938
Q:

Jack takes 20 minutes to jog around the race course one time, and 25 minutes to jog around a second time. What is his average speed in miles per hour for the whole jog if the course is 3 miles long?

 A) 6 B) 8 C) 9 D) 10