Question

As a balloon rises, its angle of elevation from a point A on level ground 140m. from the point B directly under the balloon changes from 30 degrees to 60 degrees . How far does the balloon rise during this period?

Answer 1

As a balloon rises, its angle of elevation from a point A on level ground 140m. from the point B directly under the balloon changes from 30 degrees to 60 degrees. How far does the balloon rise during this period?

**Solution:**

$\alpha_{1} = 30{}^{\circ}$ - lower angle;

$\alpha_{2} = 60{}^{\circ}$ - greater angle;

For the height at lower angle (right triangle ADC):

$\tan \alpha_{1} = \frac{h_{1}}{d}$

$h_1 = d \cdot \tan \alpha_1$ (1)

For the height at greater angle (right triangle BDC):

$\tan \alpha_{2} = \frac{h_{2}}{d}$

$h_2 = d \cdot \tan \alpha_2$ (2)

To find how much balloon was displaced, we need to subtract from the end position (height $h_2$) the starting position (height $h_1$):

$\Delta h = h_2 - h_1$ (3)

Substitute (2) and (1) in(3):

$$\begin{array}{l} \Delta h = h_2 - h_1 = d \cdot \tan \alpha_2 - d \cdot \tan \alpha_1 = d (\tan \alpha_2 - \tan \alpha_1) \\ = 140\,\mathrm{m} \cdot (\tan 60{}^{\circ} - \tan 30{}^{\circ}) = 161.7\,\mathrm{m} \end{array}$$

**Answer:** balloon rose during this time on a height of 161.7m