2. Mountain climbers ascending a high peak experience a decrease in available oxygen levels, $$ O $$, as they climb. The oxygen level function $$ O(h) $$ is given by: where $$ h $$ represents the elevation in metres above sea level. (i) At what altitude does the available oxygen level drop to half of its initial value? Give your answer correct to 1 decimal place. (ii) Determine the altitude above sea level at which the oxygen level reaches a critical (iii) Calculate the rate of change of available oxygen with respect to altitude. What does this rate represent in the context of climbing?

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To solve the problem, we need to analyze the function $$ O(h) $$ that describes the available oxygen levels as a function of altitude $$ h $$. However, the specific form of the function $$ O(h) $$ is not provided in your question.

Let's assume a common model for oxygen levels with altitude, which is often represented as an exponential decay function. A typical form could be:

$$
O(h) = O_0 e^{-kh}
$$

where:
- $$ O_0 $$ is the initial oxygen level at sea level,
- $$ k $$ is a positive constant that represents the rate of decrease of oxygen with altitude,
- $$ h $$ is the altitude in meters.

### (i) At what altitude does the available oxygen level drop to half of its initial value?

To find the altitude where the oxygen level drops to half of its initial value, we set up the equation:

$$
O(h) = \frac{1}{2} O_0
$$

Substituting the function into the equation gives:

$$
O_0 e^{-kh} = \frac{1}{2} O_0
$$

Dividing both sides by $$ O_0 $$ (assuming $$ O_0 \neq 0 $$):

$$
e^{-kh} = \frac{1}{2}
$$

Taking the natural logarithm of both sides:

$$
-kh = \ln\left(\frac{1}{2}\right)
$$

Solving for $$ h $$:

$$
h = -\frac{\ln\left(\frac{1}{2}\right)}{k}
$$

Now, we need to calculate this value. However, we need the value of $$ k $$ to proceed. If you provide the value of $$ k $$, I can calculate $$ h $$ for you.

### (ii) Determine the altitude above sea level at which the oxygen level reaches a critical level.

To determine the altitude at which the oxygen level reaches a critical level, we need to know the specific critical level of oxygen, which is not provided. Let's denote this critical level as $$ O_c $$. The equation would be:

$$
O(h) = O_c
$$

Substituting the function gives:

$$
O_0 e^{-kh} = O_c
$$

Dividing both sides by $$ O_0 $$:

$$
e^{-kh} = \frac{O_c}{O_0}
$$

Taking the natural logarithm:

$$
-kh = \ln\left(\frac{O_c}{O_0}\right)
$$

Solving for $$ h $$:

$$
h = -\frac{\ln\left(\frac{O_c}{O_0}\right)}{k}
$$

Again, we need the values of $$ O_c $$ and $$ k $$ to calculate $$ h $$.

### (iii) Calculate the rate of change of available oxygen with respect to altitude.

The rate of change of available oxygen with respect to altitude is given by the derivative of $$ O(h) $$:

$$
\frac{dO}{dh} = \frac{d}{dh}(O_0 e^{-kh}) = -k O_0 e^{-kh}
$$

This derivative represents how quickly the oxygen level decreases as altitude increases. In the context of climbing, a higher absolute value of this derivative indicates a more rapid decrease in available oxygen, which can affect the climbers' performance and safety.

### Summary

To provide specific numerical answers for parts (i) and (ii), please provide the values of $$ k $$ and $$ O_c $$.
To determine the altitude where oxygen levels drop to half and to a critical level, and to calculate the rate of change of oxygen with altitude, we need the specific values of the decay constant $$ k $$ and the critical oxygen level $$ O_c $$.

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