# Which Gives The Best Approximation to The Height of The Tower?

A vertical tower standing on a levelled field is mounted with a vertical flag staff of length 3m. From a point on the field, the angles of elevation of the bottom and tip of the flag staff are 30^{o} and 45^{o} respectively. Which one of the following gives the best approximation to the height of the tower?

- 3.90 m
- 4.00 m
- 4.10 m
- 4.25 m

**Answer; (3) 4. 10 m**

**Explanation:-**

In the figure-

Let AB = h metres and DA = X meters

Then, In Triangle ABD ,

tan 30 degree = AB/AD

1/sqrt.3 = h/X

X = h sqrt.3 ……………………… (1)

Again, in Triangle ADC,

tan 45 degree = AC/AD

AC = AD tan 45 degree

h + 3 = AD

h + 3 = X …………………………………………. (2)

Comparing equation (1) & (2),

Then, h sqrt. 3 = h + 3

h sqrt. 3 – h = 3

h (sqrt. 3 – 1) = 3

h = 3/(sqrt. 3 – 1)

h = 3 (sqrt.3 + 1)/(sqrt. 3 – 1) (sqrt.3 + 1)

h = 3 (sqrt. 3 + 1)/(3 – 1)

h = 3 (1.732 + 1)/2

h = 1.5 x 2.732

h = 4.098 or 4.10 (approximately)

Hence, the answer is (3) 4.10 m