Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Heights and Distances

NDA 2021 · Mathematics · Heights and Distances

A ladder 6 m long reaches a point 6 m below the top of a vertical...

A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is 75∘75^{{}^{\circ}}. What is the height of the flagstaff?

Options

  1. A12m12 m
  2. B9m9 m
  3. C(6+3)m(6+\sqrt{3}) \mathrm{m}
  4. D(6+33)m(6+3 \sqrt{3}) \mathrm{m}

Correct answer

D. (6+33)m(6+3 \sqrt{3}) \mathrm{m}

Step-by-step solution

Let ACA C is a flagstaff of length hmandBDh \mathrm{ m} and B D is a ladder of length 6 m . Point BB is below the top of the flagstaff, such that AB=6mA B=6 \mathrm{ m}. Then, ∠ADC=75∘\angle A D C=75^{{}^{\circ}}.
 From  △ACD,∠CAD+∠ADC+∠ACD=180∘∴∠CAD+75∘+90∘=180∘∴∠CAD=180∘−90∘−75∘=15∘\begin{aligned} & \text{\text{ From } } \triangle A C D, \angle C A D+\angle A D C+\angle A C D=180^{{}^{\circ}} \\ & \therefore \angle C A D+75^{{}^{\circ}}+90^{{}^{\circ}}=180^{{}^{\circ}} \\ & \therefore \angle C A D=180^{{}^{\circ}}-90^{{}^{\circ}}-75^{{}^{\circ}}=15^{{}^{\circ}}\end{aligned} As, AB=BD. ∴∠CAD=∠BDA=15∘ and ∠BDC=75∘−15∘=60∘. \begin{aligned} & \text{ As, } A B=B D \text{. } \\ & \therefore \angle C A D=\angle B D A=15^{{}^{\circ}} \text{ and } \\ & \angle B D C=75^{{}^{\circ}}-15^{{}^{\circ}}=60^{{}^{\circ}} \text{. }\end{aligned} From △BCD,sin⁡(60∘)=BCBD=BC6∴BC=6⋅sin⁡(60∘)=6×32=33\begin{aligned} & \text{ From } \triangle B C D, \sin \left(60^{{}^{\circ}}\right)=\frac{B C}{B D}=\frac{B C}{6} \\ & \therefore B C=6 \cdot \sin \left(60^{{}^{\circ}}\right)=6 \times \frac{\sqrt{3}}{2}=3 \sqrt{3}\end{aligned} Hence, height of flagstaff =(6+33)m=(6+3 \sqrt{3}) \mathrm{m}.
Practice Heights and Distances on Quantrex Academy →

More from Heights and Distances

A plane is observed to be approaching the airport. It is at a distance of 10 km from the… NDA 2026Consider the following for the two (02) items that follow: The top (M ) of a tower is… NDA 2025A man at M , standing 100 m away from the base (P ) of a chimney of height 50 m, observes… NDA 2025Consider the following for the two (02) items that follow: The top (M ) of a tower is… NDA 2025Paragraph: A flagstaff 20 m long standing on a pillar 10 m high subtends an angle ^ -1… NDA 2023Paragraph: There are two points P and Q due south of a leaning tower, which leans towards… NDA 2022Paragraph: There are two points P and Q due south of a leaning tower, which leans towards… NDA 2022A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff… NDA 2022Paragraph: There are two points P and Q due south of a leaning tower, which leans towards… NDA 2022The shadow of a tower is found to be x metre longer, when the angle of elevation of the… NDA 2021The angle of elevation of a tower of height h from a point A due South of it is x and… NDA 2019A ladder 9 m long reaches a point 9 m below the top of a vertical flagstaff. From the… NDA 2019A balloon is directly above one end of a bridge. The angle of depression of the other end… NDA 2018The top of a hill observed from the top and bottom of a building of height h is at angles… NDA 2018If a flag-staff of 6 m height placed on the top of a tower throws a shadow of 2 3 m along… NDA 2018A spherical balloon of radius r subtends an angle α at the eye of an observer, while the… NDA 2018 Full Heights and Distances list Heights and Distances JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs