Question

A scientist has discovered that the temperature in Kelvins of a layer in the earth’s atmosphere can be given by T=300(2-0.01Ln(L))
, where l is the height in kilometers of the layer to the ground. Find the temperature of a point which is 180 kilometers high. Round to the nearest tenth of a K.

Answer 1

To find the temperature \( T \) at a height \( L \) of 180 kilometers, we can use the given formula:

\[
T = 300(2 - 0.01 \ln(L))
\]

We substitute \( L = 180 \) into the equation:

\[
T = 300(2 - 0.01 \ln(180))
\]

Now, we need to calculate \( \ln(180) \):

\[
\ln(180) \approx 5.192
\]

Next, we substitute this value back into the equation:

\[
T = 300\left(2 - 0.01 \cdot 5.192\right)
\]

Calculating \( 0.01 \cdot 5.192 \):

\[
0.01 \cdot 5.192 = 0.05192
\]

Now, substitute this back into the formula:

\[
T = 300\left(2 - 0.05192\right) = 300(1.94808)
\]

Calculating \( 300 \cdot 1.94808 \):

\[
T \approx 584.424
\]

Finally, rounding to the nearest tenth of a K, we find:

\[
T \approx 584.4 \, \text{K}
\]

Thus, the temperature at a height of 180 kilometers is approximately \( \boxed{584.4} \) K.