Question

suppose a sourdough starter begins with 75 yeast colonies and after 4 days the population has grown to 109 yeast colonies.
a) write a continuous growth function $a = a_0e^{rt}$ that models the population over time. to do this you will need to solve for the rate of growth.
$a = 75e^{0.0935t}$
use $t$ as your variable and keep at least 4 decimals on the growth rate.
b) according to your model, what is the population after 7 days?
144 yeast colonies
round to the nearest whole number.
c) how long will it take for the population to reach 300 yeast colonies?
$t = \square$ days
give your answer accurate to at least 2 decimal places.
question help: message instructor

Explanation:

Step1: Identify known values

$A_0=75$, $A=109$, $t=4$

Step2: Solve for growth rate $r$

Substitute into $A=A_0e^{rt}$: $109=75e^{4r}$
Divide both sides by 75: $\frac{109}{75}=e^{4r}$
Take natural log: $\ln(\frac{109}{75})=4r$
Calculate $r=\frac{\ln(109/75)}{4}\approx0.0935$
Function: $A=75e^{0.0935t}$

Step3: Find population at $t=7$

Substitute $t=7$: $A=75e^{0.0935×7}$
Calculate exponent: $0.0935×7≈0.6545$
$e^{0.6545}≈1.924$
$A≈75×1.924≈144$

Step4: Solve for $t$ when $A=300$

$300=75e^{0.0935t}$
Divide by 75: $4=e^{0.0935t}$
Take natural log: $\ln(4)=0.0935t$
$t=\frac{\ln(4)}{0.0935}≈\frac{1.3863}{0.0935}≈14.83$

Answer:

a) $A=75e^{0.0935t}$
b) 144
c) 14.83