Answer to Question #338366 in Calculus for Jas

Question

Activity 2

Directions :Solve the logistics problem. show your solution.

The population of a certain bacteria follows the logistic growth pattern, initially, there are 10 g of bacteria present in the culture. Two hours later, the culture weighs 25 g. The maximum weight of the culture is 100g.

a.Write the corresponding logistic model for the bacterial growth?

b. What is the weight of the culture after 5 hours?

c. When will the culture's weight be 75 g?

Answer 1

A.

Exponential growth

$$P(t)=P(0)e^{kt}, k>0$$

Given $P(0)=10\ g, P(2)=25\ g, P\leq 100\ g$

$$P(t)=10e^{kt}$$ $$25=10e^{k(2)}$$ $$2k=\ln(2.5)$$ $$k=\dfrac{\ln(2.5)}{2}$$

The corresponding logistic model for the bacterial growth

$$P(t)=10(2.5)^{t/2}, P(t)\le100$$

B.

$$P(5)=10(2.5)^{5/2}$$ $$P(5)\approx98.821\ g$$

C.

$$P(t)=10(2.5)^{t/2}$$

$P(t)=75$

$$75=10(2.5)^{t/2}$$ $$\ln(2.5)\cdot(\dfrac{t}{2})=\ln(7.5)$$ $$t=2\cdot\dfrac{\ln(7.5)}{\ln(2.5)}$$ $$t\approx4.398\ h$$