AP Calculus BC 7.8 DE Models in Context Overview

Review differential-equation models by translating contextual rates, interpreting solutions and checking units or limiting behaviour.

Covered syllabus points

FUN-7.F—Interpret the meaning of a differential equation and its variables in context FUN-7.G—Determine general and particular solutions for problems involving differential equations in context

[Maximum number: 2]

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.

[Maximum number: 1]

The rate at which a rumor spreads across a campus of college students is given by $\frac{d P}{d t}=0.16(1200-P)$ , where P(t) represents the number of students who have heard the rumor after t days. If 200 students heard the rumor today (t=0), how many will have heard it by midnight the day after tomorrow (t=2) ?

320

474

726

1,015

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Covered syllabus points

7.8 Exponential Models with Differential Equations question 1

7.8 Exponential Models with Differential Equations question 2