Question

Scientists have found that the temperature in kelvins of a certain layer of the atmosphere can be given by

T=300(2−0.01ln(x)),

where x is the height in kilometers of the layer to the ground. If the temperature is 584 K, how high in kilometers is the layer in the atmosphere. Keep to four decimal places as you are solving the problem. Round your final answer to the nearest whole number.

Answer 1

To solve for the height (x) where the temperature is 584 K, we use the equation T=300(2-0.01ln(x)), solve for ln(x), and then compute x. After calculation, we find x to be approximately 206.0081, which we round to 206 kilometers.

Step-by-step explanation:

To find the height (x) in kilometers of the layer in the atmosphere at which the temperature is 584 K, we use the given temperature equation:

T=300(2−0.01ln(x)).

First, we substitute 584 for T and solve for x:

584 = 300(2 - 0.01ln(x))

Now, we divide both sides by 300 to isolate the term with the natural logarithm:

1.9467 = 2 - 0.01ln(x)

Next, subtract 2 from both sides:

−0.0533 = −0.01ln(x)

Now, divide both sides by -0.01 to isolate the natural logarithm:

ln(x) = 5.33

To find x, we exponentiate both sides using e to the power of the natural logarithm:

x = e^5.33

After calculating e^5.33, we determine that x is approximately:

x = 206.0081

Rounded to the nearest whole number, the height is 206 kilometers.