$$27 + 13k = 446$$

The California condor is the largest North American land bird. In 1987, it was declared extinct in the wild and only 27 birds remained in captivity. The population $k$ years after 1987 was 446, and $k$ satisfies the equation above. Which of the following is the best interpretation of the number 13 in this context?

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Compare our equation to the standard linear model:

Only the last answer choice refers to a change in population over time.

We can use dimensional analysis to find reasonable units for 13:

$$446 \text{ birds}=(13 \text{ units})(k \text{ years})+27 \text{ birds}$$ $$\colorbox{aqua}{\text{birds}}= (\text {units})(\colorbox{yellow}{\text{ years}}) + \colorbox{aqua}{\text{ birds}}$$

In order for this equation to make sense, we need a unit to convert years into birds.

$$\colorbox{aqua}{\text{birds}}= \left(\frac{\colorbox{aqua}{\text{birds}}}{\colorbox{yellow}{\text{years}}}\right)(\colorbox{yellow}{\text{ years}}) + \colorbox{aqua}{\text{ birds}}$$ $$\text{birds}= \left( \frac{\text{birds}}{\cancel{\text{years}}} \right) (\cancel{\text{years}}) + \text{birds}$$ $$\text{birds} = \text{birds}+ \text{birds}   \checkmark$$

Therefore, the units for $13$ must be birds per year. Only the last answer choice reflects this.