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 $B ( 0 ) = 20$. (a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning. (b) Find $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ in terms of $B$. Use $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ to explain why the graph of $B$ cannot resemble the following graph. (c) Use separation of variables to find $y = B ( t )$, the particular solution to the differential equation with initial condition $B ( 0 ) = 20$.
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 $B ( 0 ) = 20$.
(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning.
(b) Find $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ in terms of $B$. Use $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ to explain why the graph of $B$ cannot resemble the following graph.
(c) Use separation of variables to find $y = B ( t )$, the particular solution to the differential equation with initial condition $B ( 0 ) = 20$.