Suppose that a certain population obeys the logistic equation

$\frac{dy}{dt}=ry(1-\frac{y}{K})$

a$)$ If $y_0=\frac{K}{8},$ find the time $$ at which the initial population has doubled. Find the value of $$ corresponding to $r=0\mathrm{.}02$ per year. NOTE: Enter the exact answer or approximate your answer to $2$ decimal places.

$=$

$$

b$)$ If $\frac{y_0}{K}=,$ find tht time $T$ at which $\frac{y(T)}{K}=,$ where Observe that $T$ as $0$ or as $1.$ Find the value of $T$ for $r=0\mathrm{.}02$ per year, $=\frac{1}{8},$ and $=0\mathrm{.}65.$

NOTE: Enter the exact answer or approximate your answer to $2$ decimal places.

$T=$

$$

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Hint

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The general solution of a logistic equation with initial value $y(0)=y_0$ is given by

$y(t)=\frac{y_{0}K}{y_0(K-y_0)e^{-rt}}.$