$$ 27 + 13k = 446 $$
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?
Compare our equation to the standard linear model:
Condor population
$$ 446=13k+27 $$
$$ \text{slope}=\frac{\text{population}}{\text{year}} $$
Slope-intercept form
$$ y=mx+b $$
$$ \text{slope}=\frac{y}{x} $$
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.