Question

1. a population of yeast cells can double in 2 h. assume an initial population of 120 cells.

a) what is the growth rate, in percent per hour, of this colony of yeast cells? 1a
b) write an equation that can be used to determine the population p of cells at t hours. 1a
c) use your equation to determine the population after 210 min. 3a
d) approximately how many hours would it take for the population to reach 500 000 cells? 3a

Explanation:

Step1: Find the growth - rate formula

The general formula for exponential growth is $P = P_0(1 + r)^t$, where $P_0$ is the initial population, $r$ is the growth rate per time - unit, and $t$ is the number of time - units. We know that the population doubles in $t = 2$ hours. Let $P_0=a$ and $P = 2a$. Then $2a=a(1 + r)^2$.

Step2: Solve for the growth rate $r$

Divide both sides of the equation $2a=a(1 + r)^2$ by $a$ (since $a = 120
eq0$), we get $2=(1 + r)^2$. Take the square root of both sides: $1 + r=\sqrt{2}$ (we take the positive root since $r>0$ for growth), so $r=\sqrt{2}-1\approx1.414 - 1=0.414$ or $41.4\%$ per hour.

Step3: Write the population equation

The general exponential - growth equation is $P = P_0(1 + r)^t$. Substituting $P_0 = 120$ and $r=\sqrt{2}-1$, we get $P = 120(\sqrt{2})^{\frac{t}{1}}$, where $t$ is in hours.

Step4: Convert 210 minutes to hours

Since $1$ hour has $60$ minutes, $210$ minutes is $t=\frac{210}{60}=3.5$ hours.

Step5: Calculate the population after 3.5 hours

Substitute $t = 3.5$ into the equation $P = 120(\sqrt{2})^t$. $P = 120(\sqrt{2})^{3.5}=120\times2^{\frac{3.5}{2}}=120\times2^{1.75}$. Using a calculator, $2^{1.75}\approx3.364$, so $P = 120\times3.364 = 403.68\approx404$ cells.

Step6: Find the time when the population reaches 500000 cells

Set $P = 500000$ in the equation $P = 120(\sqrt{2})^t$. Then $\frac{500000}{120}=(\sqrt{2})^t$, or $\frac{12500}{3}=(\sqrt{2})^t$. Take the natural logarithm of both sides: $\ln(\frac{12500}{3})=t\ln(\sqrt{2})$. Since $\ln(\sqrt{2})=\frac{1}{2}\ln(2)\approx0.347$ and $\ln(\frac{12500}{3})\approx8.55$, then $t=\frac{\ln(\frac{12500}{3})}{\ln(\sqrt{2})}=\frac{8.55}{0.347}\approx24.64$ hours.

Answer:

a) $41.4\%$
b) $P = 120(\sqrt{2})^t$
c) $404$ cells
d) $24.64$ hours