The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function n = f(t) = a / (1 + be^-0.8t) where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 14 cells/hour. Find the values of a and b. a = 20(1+b) b = 14 According to this model, what happens to the yeast population in the long run? The yeast population will stabilize at 7 cells. The yeast population will grow without bound. The yeast population will shrink to 0 cells. The yeast population will stabilize at 80 cells. The yeast population will stabilize at 160 cells.
Added by Nichole H.
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So we can substitute these values into the function to find the value of b. 20 = f(0) = 1 + b * e^(0.8 * 0) 20 = 1 + b b = 20 - 1 = 19 Show more…
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The population of a culture of yeast cells is studied to see the effects of limited resources (food, space) on population growth. (a) Make a graph of the yeast population (measured as the total mass of yeast cells, tabulated below) versus time. Draw a best-fit smooth curve. (b) After a long time, the population approaches a maximum known as the carrying capacity. Estimate the carrying capacity for this population. (c) When the population is much smaller than the carrying capacity, the growth is expected to be exponential: $m(t)=m_{0} e^{r t}$ where $m$ is the population at any time $t, m_{0}$ is the initial population, $r$ is the intrinsic growth rate (i.e., the growth rate in the absence of limits), and $e$ is the base of natural logarithms (see Appendix A.4). To obtain a straight-line graph from this exponential relationship, we can plot the natural logarithm of $m / m_{0}$ $$ \ln \frac{m}{m_{0}}=\ln e^{n}=r t $$ Make a graph of $\ln \left(m / m_{0}\right)$ versus $t$ from $t=0$ to $t=6.0 \mathrm{h}$ and use it to estimate the intrinsic growth rate $r$ for the yeast population.
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