Chapter 21, Problem 106A

To determine

the approximate height above the lake up to which the mountain reaches.

Answer to Problem 106A

The height of the mountain from the lake is 2 km.

Explanation of Solution

Given:

The diameter of the lake is $d_y=$ 100 m.

Distance of the mountain from the lake is $d_m=$ 50 km.

Height of the person is $h_y$ = 2m.

Formula used:

From the concept of similarity of triangles, we can have,

  $\frac{h_y}{d_y}=\frac{h_m}{d_m}$  ......( (1)

Here, $h_m$ is the height of the mountain, $h_y$ is the height of the person, $d_m$ is the distance of the mountain from the lake and $d_y$ is the distance of the person from the lake.

Calculation:

Light has the property to reflect from the surface. The angle of incidence from a smooth surface is equal to the angle of reflection. The incident ray, reflected ray and normal line to the surface lie on the same plane.

Rearrange the equation (1) for $h_m$ ,

  $h_m=\left(\frac{h_y}{d_y}\right)d_m$  ......(2)

Substitute the values of $h_y$ , $d_m$ and $d_y$ in the equation (2),

  $\begin{array}{l}{h}_m=\left(\frac{2\text{m}}{50\text{m}}\right)\left(50\text{km}\right)\\ h_m=\left(\frac{2\text{m}}{50\text{m}}\right)\left(50\times {10}^3\text{m}\right)\\ h_m=2\times {10}^3\text{m}\\ \boxed{h_m=2\text{km}}\end{array}$

Conclusion:

Hence, the height of the mountain from the lake is 2 km.