Applications Of Trigonometry(10th Grade > Mathematics ) Questions and answers for exam Preparation

10th Grade > Mathematics

APPLICATIONS OF TRIGONOMETRY MCQs

Total Questions : 58 | Page 6 of 6 pages

The string of a kite is 100m long and it makes an angle of $60^{\circ}$ with the horizontal. Find the height of the kite from the ground, assuming that there is no slack in the string.

100 $\sqrt{3}$
$\frac{200}{\sqrt{3}}$
50 $\sqrt{3}$
$\frac{100}{\sqrt{3}}$

Discuss Question

Answer: Option C. -> 50

\sqrt{3}

:
C

We know, angle of elevation = $60^{\circ}$
$\Rightarrow$ $sin$ $θ$ = $\frac{p e r p e n d i c u l a r}{h y p o t e n u s e}$
$\Rightarrow$ $sin$ $60^{\circ}$ = $\frac{\sqrt{3}}{2}$ = $\frac{h}{100}$
$\Rightarrow$ h = 50 $\sqrt{3} m$ .

Question 52.

The angle of elevation of the top of a tower from a point on the ground, which is 30m away from the foot of the tower is $30^{\circ}$ . Find the height of the tower.

10 $\sqrt{3}$ m
20 $\sqrt{3}$ m
30 $\sqrt{3}$ m
20 m

Discuss Question

Answer: Option A. -> 10

\sqrt{3}

m
:
A

Angle of elevation = $θ$ = $30^{\circ}$
$\Rightarrow$ tan $θ$ = $\frac{P e r p e n d i c u l a r}{B a s e}$
$\Rightarrow$ tan $θ$ = $\frac{h}{30}$
$\Rightarrow$ tan $30^{\circ}$ = $\frac{1}{\sqrt{3}}$
$\Rightarrow$ $\frac{h}{30}$ = $\frac{1}{\sqrt{3}}$
Therefore, $h = 10 \sqrt{3} m$ .

Question 53.

Two towers A and B are standing at some distance apart. From the top of tower A, the angle of depression of the foot of tower B is found to be $30^{\circ}$ . From the top of tower B, the angle of depression of the foot of tower A is found to be $60^{\circ}$ . If the height of tower B is ‘h’ m then the height of tower A in terms of ‘h’ is _____ m

$\frac{h}{2}$ m
$\frac{h}{3}$ m
$\sqrt{3} h$ m
$\frac{h}{\sqrt{3}}$ m

Discuss Question

Answer: Option B. ->

\frac{h}{3}

m
:
B

Let the height of tower A be = AB = H.

And the height of tower B = CD = h

In triangle ABC

tan $30^{\circ}$ = $\frac{A B}{A C}$ = $\frac{H}{A C}$ ...... (1)

In triangle ADC

tan $60^{\circ}$ = $\frac{C D}{A C}$ = $\frac{h}{A C}$ ...... (2)

Dividing (1) by (2) we get, $\frac{t a n 30^{\circ}}{t a n 60^{\circ}}$ = $\frac{H}{h}$

H= $\frac{h}{3}$

Question 54.

Find the angle of elevation of the sun when the shadow of a 10m long pole is 10 $\sqrt{3}$ m.

$30^{\circ}$
$60^{\circ}$
$45^{\circ}$
$15^{\circ}$

Discuss Question

Answer: Option A. ->

30^{\circ}

:
A

Let, angle of elevation of sun = $θ$
$\Rightarrow$ Tan $θ$ = $\frac{P e r p e n d i c u l a r}{b a s e}$ = $\frac{10}{10 \sqrt{3}}$
$\Rightarrow$ Tan $θ$ = $\frac{1}{\sqrt{3}}$
$\Rightarrow$ $θ$ = $30^{\circ}$

Question 55.

A circus artist is climbing a 20m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. The height of the pole is

___
m if the angle made by the rope with the ground level is

30^{\circ}

Discuss Question

Answer: Option A. ->

30^{\circ}

Angle of elevation $θ$ = $30^{\circ}$

$\Rightarrow$ Sin $θ$ = $\frac{p e r p e n d i c u l a r}{h y p o t e n u s e}$

$\Rightarrow$ Sin $30^{\circ}$ = $\frac{1}{2}$ = $\frac{h e i g h t o f p o l e}{20}$

Thus, height of pole = 10 m.

Question 56.

The angle of elevation of the top of a tree from a point A on the ground is $60^{\circ}$ . On walking 20 m away from point A, to a point B, the angle of elevation changes to $30^{\circ}$ . Find the height of the tree.

$10 \sqrt{3}$ m
$20 \sqrt{3}$ m
$30 \sqrt{3}$ m
$40 \sqrt{3}$ m

Discuss Question

Answer: Option A. ->

10 \sqrt{3}

m
:
A

In triangle ACD,

$tan θ = \frac{D C}{C A} = \frac{h}{x}$

$\Rightarrow tan 60^{\circ} = \sqrt{3} = \frac{h}{x}$ ..................... 1

In triangle CDB,

$\Rightarrow tan 30^{\circ} = \frac{C D}{C B}$

$\Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{x + 20}$ ....................... 2
Using (1) and (2), we get

$\Rightarrow \frac{1}{\sqrt{3}} = \frac{x \sqrt{3}}{x + 20}$
$\Rightarrow x + 20 = 3 x$
$\Rightarrow 2 x = 20$
$\Rightarrow x = 10$
$∴ h = x \sqrt{3} = 10 \sqrt{3}$
Thus, the height of the tree is $10 \sqrt{3} m$ .

Question 57.

The tops of two poles of height 14 m and 20 m are connected by a wire which makes an angle of 30° with the horizontal. The length of the wire is _____m.

Discuss Question

Answer: Option D. -> 12m
:
D

In triangle ABC,

BC = y = 20-14 = 6m;

Let AB = $x$

$\Rightarrow$ Sin $30^{\circ}$ = $\frac{B C}{A B}$

$\Rightarrow$ $\frac{1}{2}$ = $\frac{6}{x}$

$\Rightarrow$ $x$ = 12

Thus, the length of the string is 12 m.

Question 58.

From the top of a cliff 25m high the angle of elevation of a tower is found to be equal to the angle of depression to the foot of the tower. The height of the tower is ___.