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 Homework Statement

An electron of the energy $$E_0 = 10TeV$$ enters the earth's atmosphere and releases
a particle shower. For the sake of simplicity, assume that the atmosphere is isothermal (T = 273 K),
the pressure at the ground is 100 kPa and the gravitational acceleration does not change with altitude.
The first interaction of the electron takes place when it comes from space and enters a radiation length deep into the atmosphere. Calculate the height of the point of the first interaction of the electron above the ground (in km).
 Homework Equations

$$\rho = \rho_0\cdot e^{\frac{h_1h_0}{RT}\cdot m_{mol}g}$$
with R the gas constant
The radiaton length for air is about $$X_0 = 30420cm$$.
This is the length at which the electron has decreased to 1/e of it´ s initially value.
I also know that the maximal value of interactions for a specific energy is given by $$ n_{max} = \frac{ln(\frac{E_0}{E_c})}{ln(2)} $$, where E_c is the critical energy, which I calculated to $$ E_c = 79,59eV$$.
So I got $$n_{max} = 13,61$$.
But this this still doesn´ t really help me to determine the heigth of the first point of interaction.
Maybe someone can help an give me a hint.
This is the length at which the electron has decreased to 1/e of it´ s initially value.
I also know that the maximal value of interactions for a specific energy is given by $$ n_{max} = \frac{ln(\frac{E_0}{E_c})}{ln(2)} $$, where E_c is the critical energy, which I calculated to $$ E_c = 79,59eV$$.
So I got $$n_{max} = 13,61$$.
But this this still doesn´ t really help me to determine the heigth of the first point of interaction.
Maybe someone can help an give me a hint.
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