AP Calculus BC 7.8: DE Models in Context

Practice AP Calculus BC questions on interpreting a differential equation as a rate model and comparing rates for different quantities.

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The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t=0, when the bird is first weighed, its weight is 20 grams. If B(t) is the weight of the bird, in grams, at time t days after it is first weighed, then

\frac{d B}{d t}=\frac{1}{5}(100-B) .

Let y=B(t) be the solution to the differential equation above with initial condition B(0)=20.

Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning.

$\left.\frac{d B}{d t}\right|_{B=40}=\frac{1}{5}(60)=12$

\left.\frac{d B}{d t}\right|_{B=70}=\frac{1}{5}(30)=6

Because $\left.\frac{d B}{d t}\right|_{B=40}>\left.\frac{d B}{d t}\right|_{B=70}$ , the bird is gaining weight faster when it weighs 40 grams.

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