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 $$\frac{dB}{dt} = \frac{1}{5}(100 - B).$$ 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 $\frac{d^{2}B}{dt^{2}}$ in terms of $B$. Use $\frac{d^{2}B}{dt^{2}}$ 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
$$\frac{dB}{dt} = \frac{1}{5}(100 - B).$$
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 $\frac{d^{2}B}{dt^{2}}$ in terms of $B$. Use $\frac{d^{2}B}{dt^{2}}$ 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 $B(0) = 20$.