Chapter 9 · Applications of Trigonometry

NCERT Class 10 Maths Exercise 9.1

All 15 questions of NCERT Class 10 Maths Exercise 9.1 (Applications of Trigonometry), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   A circus artist is climbing a 20 ​m long rope, which is tightly stretched and tied ​from ​the top of a vertical pole to ​the ground. Find the height of the pole, if the angle made by the rope with the ground level is $30^{\circ }$ ​(see Fig. 9.11).

   Show answer

   Height of the pole $=10$ m.

   Step-by-step solution →
2. Q2
   A tree breaks due to storm ​and the broken part ​bends so that the ​top of the tree touches the ground making an angle $30^{\circ }$ ​with it. The distance ​between the foot of the tree to the ​point where the top touches the ground is 8 m. Find the ​height of ​the tree.
   Show answer

   Height of the tree $=8\sqrt{3}$ m $\approx 13.86$ m.

   Step-by-step solution →
3. Q3
   A contractor plans to install two slides for the children to play in a park. For the children ​below the age of $5$ years, she ​prefers to have a ​slide whose top is at a height of $1.5$ m, and is inclined at ​an angle of $30^{\circ }$ to the ground, whereas for elder children, she wants to have a steep slide at a height of $3$ m, and inclined at an angle of $60^{\circ }$ to the ground. What should be the length ​of the ​slide in each case?

   1. (i)For children below 5 years: a slide with its top at ​a height of $1.5$ m, inclined at $30^{\circ }$ to the ground. Find the length of this slide.
   2. (ii)For elder children: a steeper ​slide with its top at a height of $3$ m, inclined at $60^{\circ }$ to the ground. Find the length of this slide.
   Show answer

   (i) Slide for younger children $=3$ m. (ii) Slide for elder children $=2\sqrt{3}$ m $\approx 3.46$ m.

   Step-by-step solution →
4. Q4
   The angle of elevation of the top of a tower from a point on the ​ground, which is $30$ m away from the foot ​of ​the ​tower, ​is $30^{\circ }$. Find the ​height of ​the tower.
   Show answer

   Height of the tower $=10\sqrt{3}$ m $\approx 17.32$ m.

   Step-by-step solution →
5. Q5
   A kite is flying at ​a height ​of $60$ m above the ​ground. The ​string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with ​the ground is $60^{\circ }$. Find ​the length of ​the string, assuming that there is ​no slack in the string.
   Show answer

   Length of the string $=40\sqrt{3}$ m $\approx 69.28$ m.

   Step-by-step solution →
6. Q6
   A 1.5 m tall boy ​is standing at some distance from ​a ​30 m tall ​building. The angle of elevation from his eyes to the top ​of the building increases from $30^{\circ }$ to $60^{\circ }$ as ​he walks towards the ​building. Find the ​distance he walked towards the building.
   Show answer

   Distance walked $=19\sqrt{3}$ m $\approx 32.91$ m.

   Step-by-step solution →
7. Q7
   From ​a point ​on the ground, the angles of elevation of the bottom and the top of a transmission ​tower fixed at the top of a $20$ ​m high building are $45^{\circ }$ and $60^{\circ }$ respectively. ​Find the height of ​the ​tower.
   Show answer

   Height of the tower $=20(\sqrt{3}-1)$ m $\approx 14.64$ m.

   Step-by-step solution →
8. Q8
   A statue, $1.6$ ​m tall, stands on the top of a pedestal. From a point on the ​ground, the angle of ​elevation of the top of the statue is $60^{\circ }$ ​and from the same point the ​angle of elevation of the top of the pedestal is $45^{\circ }$. ​Find the height of ​the pedestal.
   Show answer

   Height of the pedestal $=0.8(\sqrt{3}+1)$ m $\approx 2.19$ m.

   Step-by-step solution →
9. Q9
   The angle of elevation of the top of a ​building from the foot of the ​tower is $30^{\circ }$ and the angle of elevation of the top ​of the tower from the foot of the building ​is $60^{\circ }$. If the tower is $50$ m high, find ​the height of the building.
   Show answer

   Height of the building $=\frac{50}{3}$ m $\approx 16.67$ m.

   Step-by-step solution →
10. Q10
    Two poles of equal heights are standing opposite ​each other on ​either ​side of ​the road, which is $80$ m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^{\circ }$ and $30^{\circ }$, respectively. Find ​the height of the poles and the distances of the point ​from the poles.

    1. (i)Find the height of ​the poles (both are equal).
    2. (ii)Find the distances of the point from each of the two poles.
    Show answer

    (i) Height of each pole $=20\sqrt{3}$ m $\approx 34.64$ m. (ii) The point is 20 m from the $60^{\circ }$ pole and 60 m from the $30^{\circ }$ pole.

    Step-by-step solution →
11. Q11
    A ​TV tower stands vertically on a ​bank of ​a canal. From a point on the ​other bank directly opposite the tower, the angle of elevation of the top ​of the tower is $60^{\circ }$. From another point $20$ m away from this point on the line joining ​this point to the ​foot of the tower, the angle of elevation of the top of the tower ​is $30^{\circ }$ (see Fig. 9.12). Find the height of the tower and the width of the canal.

    1. (i)Find the height of the TV tower.
    2. (ii)Find the width of the canal.

    Show answer

    (i) Height of the tower $=10\sqrt{3}$ m $\approx 17.32$ m. (ii) Width of the canal $=10$ m.

    Step-by-step solution →
12. Q12
    From the top of ​a $7$ m ​high ​building, the ​angle of elevation of the top of ​a cable ​tower is $60^{\circ }$ and the angle of depression ​of its foot ​is $45^{\circ }$. Determine the height of the tower.
    Show answer

    Height of the tower $=7(\sqrt{3}+1)$ m $\approx 19.12$ m.

    Step-by-step solution →
13. Q13
    As ​observed from the top of a 75 m high lighthouse from the sea-level, the ​angles of depression of two ships are $30^{\circ }$ and $45^{\circ }$. If one ship is exactly ​behind the other on the ​same side of the lighthouse, ​find ​the distance between the ​two ships.
    Show answer

    Distance between the ships $=75(\sqrt{3}-1)$ m $\approx 54.9$ m.

    Step-by-step solution →
14. Q14
    A $1.2$ m tall girl spots a ​balloon moving with the wind in a horizontal line at a height of $88.2$ m from the ground. The angle ​of elevation of the balloon from the eyes of the girl at any instant ​is $60^{\circ }$. After some time, the angle of ​elevation ​reduces ​to $30^{\circ }$ (see Fig. 9.13). ​Find the distance travelled by the ​balloon during the interval.

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    Distance travelled by the balloon $=58\sqrt{3}$ m $\approx 100.46$ m.

    Step-by-step solution →
15. Q15
    A straight highway ​leads to the foot of a tower. A man standing at the top ​of ​the tower observes a car at an angle of depression of $30^{\circ }$, which is approaching the ​foot of ​the tower with a uniform speed. Six seconds later, the angle of depression ​of the car is found to be $60^{\circ }$. ​Find the time taken by the car ​to reach the foot of the tower from this point.
    Show answer

    Time to reach the foot of the tower $=3$ seconds.

    Step-by-step solution →