4. a population of yeast cells can double in 2 h. assume an initial population of 120 cells. a) what is the…

Explanation:

Step1: Find the growth - rate formula

The general formula for exponential growth is $P = P_0(1 + r)^t$, where $P$ is the final population, $P_0$ is the initial population, $r$ is the growth rate per time - unit, and $t$ is the number of time - units. We know that when $t = 2$ hours, $P = 2P_0$. Let $P_0=120$, then $2P_0=P_0(1 + r)^2$. Divide both sides by $P_0$ (since $P_0\neq0$), we get $2=(1 + r)^2$.

Step2: Solve for $r$

Take the square root of both sides: $1 + r=\sqrt{2}$ (we take the positive root since $r$ represents a growth rate). Then $r=\sqrt{2}-1\approx1.414 - 1=0.414$. To convert to a percentage, we multiply by 100, so $r = 41.4%$ per hour.

Step3: Write the population equation

Using the general form $P = P_0(1 + r)^t$, with $P_0 = 120$ and $r=0.414$, the equation is $P = 120(1.414)^t$.

Step4: Convert time to hours

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

Step5: Calculate the population

Substitute $t = 3.5$ into the equation $P = 120(1.414)^t$. So $P = 120\times(1.414)^{3.5}$. First, $(1.414)^{3.5}=1.414^3\times1.414^{0.5}$. $1.414^3=1.414\times1.414\times1.414\approx2.828$ and $1.414^{0.5}=\sqrt{1.414}\approx1.19$. Then $(1.414)^{3.5}\approx2.828\times1.19 = 3.365$. So $P=120\times3.365 = 403.8\approx404$ cells.

Answer:

a) $41.4%$ b) $P = 120(1.414)^t$ c) $404$ cells