Answer
The height of the taller building is 146 m
Work Step by Step
We can convert the angle of depression to degrees:
$\theta_d = 14^{\circ}10' = (14+\frac{10}{60})^{\circ} = 14.17^{\circ}$
We can use the height $h_1$ of the shorter building to find the horizontal distance $d$ between the two buildings:
$\frac{h_1}{d} = tan~\theta_d$
$d = \frac{h_1}{tan~\theta_d}$
$d = \frac{28.0~m}{tan~(14.17^{\circ})}$
$d = 110.9~m$
We can convert the angle of elevation to degrees:
$\theta_e = 46^{\circ}40' = (46+\frac{40}{60})^{\circ} = 46.67^{\circ}$
We can use the distance $d$ between the buildings to find the additional height $h_2$ of the taller building:
$\frac{h_2}{d} = tan~\theta_e$
$h_2 = (d)~tan~\theta_e$
$h_2 = (110.9~m)~tan~(46.67^{\circ})$
$h_2 = 118~m$
The total height $h$ of the taller building is $h_1+h_2$:
$h = h_1+h_2$
$h = 28.0~m+118~m$
$h = 146~m$
The height of the taller building is 146 m
