In the given diagram, AB is a vertical tower 100 m away from the foot of a 30 storied building CD. The angles of depression from the point C and E, (E being the mid-point of CD), are 35° and 14° respectively. (Use mathematical table for the required values rounded off correct to two places of decimals only.)

Find the height of the:

(a) tower AB

(b) building CD

From figure,

Let height of the building CD be H meters and height of tower AB be h meters.

Given,

BD = 100 m

E is the midpoint of CD

∴ CE = $\dfrac{CD}{2} = \dfrac{H}{2}$

From figure,

⇒ ∠CAP = ∠XCA = 35° [Alternate angles are equal]

⇒ ∠EAP = ∠YEA = 14° [Alternate angles are equal]

From figure,

PD = AB = h

AP = BD = 100 m

In ΔACP,

$\Rightarrow \text{tan 35°} = \dfrac{\text{Perpendicular}}{\text{Base}}\\[1em] \Rightarrow \text{tan 35°} = \dfrac{CP}{AP} \\[1em] \Rightarrow \text{tan 35°} = \dfrac{CD - PD}{AP} \\[1em] \Rightarrow 0.70 = \dfrac{H - h}{100} \\[1em] \Rightarrow 0.70 × 100 = H - h \\[1em] \Rightarrow H - h = 70 …..(1)$

In ΔAEP,

$\Rightarrow \text{tan 14°} = \dfrac{\text{Perpendicular}}{\text{Base}} \\[1em] \Rightarrow \text{tan 14°} = \dfrac{EP}{AP}\\[1em] \Rightarrow \text{tan 14°} = \dfrac{ED - PD}{AP} \\[1em] \Rightarrow 0.25 = \dfrac{\dfrac{H}{2} - h}{100} \\[1em] \Rightarrow \dfrac{H}{2} - h = 0.25 × 100 \\[1em] \Rightarrow \dfrac{H}{2} - h = 25 …..(2)$

Subtracting equation (2) from (1), we get:

$\Rightarrow H - h - \Big({\dfrac{H}{2} - h}\Big) = 70 - 25 \\[1em] \Rightarrow H - h - {\dfrac{H}{2} + h} = 45 \\[1em] \Rightarrow H - \dfrac{H}{2} = 45 \\[1em] \Rightarrow \dfrac{H}{2} = 45 \\[1em] \Rightarrow H = 45 \times 2 = 90 \text{ m}.$

⇒ H - h = 70

⇒ 90 - h = 70

⇒ h = 90 - 70

⇒ h = 20 m.

Hence, height of the tower AB = 20 m and height of the building CD = 90 m.