Question

At a particular time, when the sun’s altitude is 30°, the length of the shadow of a vertical tower is 45 m. Calculate:

1. the height of the tower.
2. the length of the shadow of the same tower, when the sun’s altitude is:

1. 45°
2. 60°

Answer 1

Shadow of the tower = 45 m and angle of elevation = 30°

i. Let AB be the tower and BC is its shadow.

∴ CB = 45 m.

Now in right ΔABC,

`tan theta = (AB)/(BC)`

`\implies tan 30^circ = (AB)/45`

`\implies 1/sqrt(3) = (AB)/45`

`\implies AB = 45/sqrt(3)m`.

∴ `AB = (45 xx sqrt(3))/(sqrt(3) xx sqrt(3))`

= `(45sqrt(3))/3`

= `15sqrt(3)  m`.

= 15 (1.732)

= 25.980

= 25.98 m

ii. In second case,

a. Angle of elevation = 45°

And height of tower = 25.98 m

or `15sqrt(3)  m`

`tan 45^circ = (AB)/(DB)`

`\implies (AB)/(DB) = 1`

∴ DB = AB = 25.98 m.

b. Angle of elevation = 60°

And height of tower = 25.98 m

or `15sqrt(3)  m`.

Let shadow of the tower DB = x m

∴ `tan 60^circ = (AB)/(DB)`

`\implies sqrt(3) = (15sqrt(3))/(DB)`

`\implies DB = (15sqrt(3))/sqrt(3) = 15  m`.

Hence length of shadow = 15 m.