Chapter 1, Problem 45GP

To determine

To Estimate: The diameter of the moon.

Answer to Problem 45GP

  $D=2.280\text{ km}$

Explanation of Solution

Given info:

Distance of moon from the earth, $R=3.8\times {10}^5\text{ km = 3}\text{.8}\times {\text{10}}^8\text{ m}$

Estimated diameter of the pencil, $d=6\text{ mm = 6}\text{.0}\times {\text{10}}^{-3}\text{ m}$

Estimated distance of pencil from the eye, $r=1.42\text{ arm-length = 1 m}$

Formula used:

The condition of similarity of two triangles:

  $\frac{D}{R}=\frac{d}{r}$

Calculation:

  

Triangle OAB and OCE are similar triangle, therefore:

  $\begin{array}{c}\frac{OA}{OC}=\frac{AB}{CE}\\ \frac{r}{R}=\frac{d}{D}\end{array}$

The diameter of the moon:

  $\begin{array}{c}D=\frac{R}{r}d\end{array}$

    $\begin{array}{c}=\frac{3.8\times {10}^8}{1}\times 6.0\times {10}^{-3}\end{array}$

    $\begin{array}{c}=22.8\times {10}^5\text{ m}\end{array}$

    $\begin{array}{c}\text{=2280 km}\end{array}$

Conclusion:

Hence,estimated diameter of the moon: $\text{2280 km}$