Question

sample question for logen01-estimation
question 1
make an order of magnitude estimate of the mass of a mountain. a mountain can be approximated as a cone with volume ( v = \frac{pi r^2 h}{3} ) where r is the radius of the base and h is the height.

Explanation:

Step1: Estimate dimensions of a mountain

A typical mountain might have a height \( h \approx 10^4 \, \text{m} \) (10 kilometers) and a base radius \( r \approx 10^4 \, \text{m} \) (assuming a roughly circular base with diameter around 20 kilometers, so radius 10 kilometers).

Step2: Calculate volume of the cone

Using the volume formula for a cone \( V=\frac{\pi r^{2}h}{3} \), substitute \( r = 10^{4}\, \text{m} \) and \( h=10^{4}\, \text{m} \):
\[

$$\begin{align*} V&=\frac{\pi\times(10^{4})^{2}\times10^{4}}{3}\\ &=\frac{\pi\times 10^{8}\times 10^{4}}{3}\\ &=\frac{\pi\times 10^{12}}{3}\\ &\approx 10^{12}\, \text{m}^3 \quad (\text{since } \pi/3\approx 1) \end{align*}$$

\]

Step3: Estimate density of mountain material

Mountain rocks (like granite) have a density \(
ho\approx 3\times 10^{3}\, \text{kg/m}^3 \).

Step4: Calculate mass using \( m =

ho V \)
\[

$$\begin{align*} m&= ho V\\ &=3\times 10^{3}\, \text{kg/m}^3\times 10^{12}\, \text{m}^3\\ &= 3\times 10^{15}\, \text{kg} \end{align*}$$

\]
The order of magnitude is determined by the power of 10, so the order of magnitude of the mass is \( 10^{16}\, \text{kg} \) (since \( 3\times 10^{15}\approx 10^{16} \) when considering order of magnitude, we round to the nearest power of 10).

Answer:

The order of magnitude estimate of the mass of a mountain is \( \boldsymbol{10^{16}\, \text{kg}} \) (or a similar power of 10 depending on more precise estimates of dimensions, but this is a typical order of magnitude estimate).