Let c be a positive number. A differential equation of the form where k is a positive constant is called a doomsday equation because the exponent in the expression is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term . If such rabbits breed initially and the warren has rabbits after months, then when is doomsday?

Question

Let c be a positive number. A differential equation of the form  where k is a positive constant is called a doomsday equation because the exponent in the expression  is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term  . If  such rabbits breed initially and the warren has  rabbits after  months, then when is doomsday?

Quizplus · Accepted Answer

The given growth term for rabbits is $y'=ky^2$, where $y$ is the number of rabbits and $k$ is a positive constant. This is of the form given in the question: $y' = k\cdot y^{\frac{11}{5}}$.

We are given that there are $y(0) = 11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11$ rabbits initially and $y(\Delta t) = 11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11$ rabbits after $\Delta t = 11eaa3e5_5528_fff1_9f8f_cf2a8d302a89_TB5971_11$ months.

Using separation of variables, we have:

$$\frac{dy}{dt} = ky^{\frac{11}{5}}$$

$$\int \frac{dy}{y^{\frac{11}{5}}} = \int k\ dt$$

$$-\frac{5}{6} y^{-\frac{6}{5}} = kt+C$$

Using the initial condition $y(0) = 11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11$:

$$-\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}} = k\cdot 0+C$$

Simplifying this expression, we get:

$$C = -\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Plugging this value of $C$ back into our equation:

$$-\frac{5}{6} y^{-\frac{6}{5}} = kt-\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Using the second initial condition $y(\Delta t) = 11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11$:

$$-\frac{5}{6} (11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11)^{-\frac{6}{5}} = k\cdot \Delta t-\frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

We want to find the value of $\Delta t$ for which $y$ becomes infinite (i.e. doomsday), so we set $y$ to infinity:

$$-\frac{5}{6} \cdot \infty^{-\frac{6}{5}} = k\cdot t - \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Simplifying this expression, we get:

$$-\frac{5}{6} \cdot 0 = k\cdot \Delta t - \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

$$k\cdot \Delta t = \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Plugging in the given values, we get:

$$k\cdot \Delta t = \frac{5}{6} (11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11)^{-\frac{6}{5}} - \frac{5}{6} (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}$$

Simplifying this expression using a common denominator, we get:

$$k\cdot \Delta t = \frac{5}{6} \cdot \frac{(11eaa3e5_5528_fff0_9f8f_773f824a7697_TB5971_11)^{-\frac{6}{5}} - (11eaa3e5_5528_ffee_9f8f_b3e6993ba9a5_TB5971_11)^{-\frac{6}{5}}}{(11eaa3e5_5528_fff1_9f8f_cf2a8d302a89_TB5971_11) - 0}$$

Using a calculator or Wolfram Alpha, we can find that the value of $\Delta t$ that satisfies this equation is approximately $11eaa3e5_5529_2705_9f8f_b9c563b683c4_TB5971_11$, so the answer is $\boxed{	extbf{(D)}}$.

Let C Be a Positive Number