Let's go through each part of this problem.

The given formula for the average level of carbon dioxide in the air is:

$q(t) = 0.05t^2 + 0.1t + 3.4$

where $q(t)$ represents the CO$_2$ level in parts per million, and $t$ is the time in years from now.

Part (a)

To find the rate at which the CO$_2$ level is changing with respect to time one year from now, we need to calculate the derivative $q'(t)$ and evaluate it at $t = 1$.

1. Differentiate $q(t)$ with respect to $t$: $q'(t) = \frac{d}{dt} (0.05t^2 + 0.1t + 3.4)$ $q'(t) = 2 \cdot 0.05 \cdot t + 0.1 = 0.1t + 0.1$

2. Substitute $t = 1$ into $q'(t)$: $q'(1) = 0.1 \cdot 1 + 0.1 = 0.1 + 0.1 = 0.2$

So, the rate at which the CO$_2$ level is changing one year from now is $0.2$ parts per million per year.

Part (b)

To find how much the CO$_2$ level changes in the first year, we need to evaluate $q(1) - q(0)$.

1. Calculate $q(1)$: $q(1) = 0.05 \cdot (1)^2 + 0.1 \cdot 1 + 3.4 = 0.05 + 0.1 + 3.4 = 3.55$

2. Calculate $q(0)$: $q(0) = 0.05 \cdot (0)^2 + 0.1 \cdot 0 + 3.4 = 3.4$

3. Find the change in CO$_2$ level: $q(1) - q(0) = 3.55 - 3.4 = 0.15$

Thus, the CO$_2$ level increases by $0.15$ parts per million in the first year.

Part (c)

To determine how much the CO$_2$ level changes over the next (second) year, we need to calculate $q(2) - q(1)$.

1. Calculate $q(2)$: $q(2) = 0.05 \cdot (2)^2 + 0.1 \cdot 2 + 3.4 = 0.05 \cdot 4 + 0.2 + 3.4 = 0.2 + 0.2 + 3.4 = 3.8$

2. Find the change in CO$_2$ level: $q(2) - q(1) = 3.8 - 3.55 = 0.25$

Therefore, the CO$_2$ level increases by $0.25$ parts per million over the second year.

Summary of Answers

• (a) The rate of change in CO$_2$ level one year from now is $0.2$ parts per million per year.
• (b) The CO$_2$ level increases by $0.15$ parts per million in the first year.
• (c) The CO$_2$ level increases by $0.25$ parts per million over the second year.

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions that could expand your understanding:

1. How would the rate of change vary if the function $q(t)$ had a different coefficient for $t^2$?
2. How can we interpret the second derivative $q''(t)$ in terms of CO$_2$ concentration change over time?
3. What would happen to the CO$_2$ levels if the initial concentration ($3.4$) were different?
4. How could we predict the CO$_2$ level for a future year, say $t = 5$?
5. How does the constant term $0.1$ in the derivative affect the rate of change over time?

Tip: Differentiating a quadratic function $ax^2 + bx + c$ always results in a linear function $2ax + b$, which can make it easier to analyze rate of change patterns.