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Mathematics · Some Applications of Trigonometry · NCERT Examples

Example 5

A complete, independently verified solution for NCERT Example 5.

Mathematics · Chapter 9 · NCERT Examples

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NCERT Example

Example 5

shadows · altitude of sun

The shadow of a tower is 40 m longer when the Sun's altitude is 30° than when it is 60°. Find the height of the tower.

Tower AB casting shadows BC and BD when the Sun's altitudes are 60 and 30 degrees.
Figure 9-8 from the supplied NCERT chapter.

Solution

Board-exam working:

Let AB = h m and BC = x m. Then BD = x + 40 m.

Using triangle ABC:

tan⁡60∘=ABBC\tan60^\circ=\frac{AB}{BC}
3=hx\sqrt3=\frac{h}{x}
h=x3(1)h=x\sqrt3\qquad\text{(1)}

Using triangle ABD:

tan⁡30∘=ABBD\tan30^\circ=\frac{AB}{BD}
13=hx+40\frac1{\sqrt3}=\frac{h}{x+40}
h=x+403(2)h=\frac{x+40}{\sqrt3}\qquad\text{(2)}

Equating (1) and (2):

x3=x+403x\sqrt3=\frac{x+40}{\sqrt3}
3x=x+403x=x+40
x=20x=20
h=203 mh=20\sqrt3\text{ m}

Final answer

The height of the tower is 20√3 m.