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Mathematics · Some Applications of Trigonometry · NCERT Examples

Example 6

A complete, independently verified solution for NCERT Example 6.

Mathematics · Chapter 9 · NCERT Examples

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NCERT Example

Example 6

angles of depression · two buildings

The angles of depression of the top and bottom of an 8 m building from the top of a multi-storeyed building are 30° and 45°, respectively. Find the height of the multi-storeyed building and the distance between the buildings.

An 8 metre building AB and multi-storeyed building PC with 30 and 45 degree angles of depression.
Figure 9-9 from the supplied NCERT chapter.

Solution

Board-exam working:

Let PD = x m. Since AB = DC = 8 m, the total height PC = x + 8 m.

In triangle PBD:

tan⁡30∘=PDBD\tan30^\circ=\frac{PD}{BD}
13=xBD\frac1{\sqrt3}=\frac{x}{BD}
BD=x3(1)BD=x\sqrt3\qquad\text{(1)}

In triangle PAC:

tan⁡45∘=PCAC=1\tan45^\circ=\frac{PC}{AC}=1
AC=PC=x+8(2)AC=PC=x+8\qquad\text{(2)}

The distance between the buildings is the same, so AC = BD.

x+8=x3x+8=x\sqrt3
x(3−1)=8x(\sqrt3-1)=8
x=83−1=4(3+1)x=\frac8{\sqrt3-1}=4(\sqrt3+1)
PC=x+8=4(3+3) mPC=x+8=4(\sqrt3+3)\text{ m}
AC=4(3+3) mAC=4(\sqrt3+3)\text{ m}

Final answer

Height of the multi-storeyed building = 4(√3 + 3) m; distance between the buildings = 4(√3 + 3) m.