Calculus problem - Mental creativity and flexability needed

• zaboda42
In summary, the spread of a disease in a population is modeled by a differential equation with the general solution P(t) = 30Ae^(0.9t)/[1 + Ae^(0.9t)]. Given that 5000 people have the disease when first noticed, the particular solution is P(t) = 6e^(0.9t)/[1 + 0.2e^(0.9t)]. It is estimated that eventually, 30,000 people will have the disease. This is based on the limit of P(t) as t approaches infinity.
zaboda42
The spread of a disease in a population is modeled by the differential equation dP/dt=.03P(30-P) where p is less than or equal to 30 is measured in thousands of people and t is greater or equal to 0 is measured in weeks.

a) Find the general solution of the differential equation.
b) Find the particular solution of the differential equation given that 5000 people have the disease when first noticed, that is, when t=0, P=5.
c) Estimate how many people will eventually have the disease. Explain your answer.

Here's what i did:(a)
dP/[P(30 - P)] = 0.03 dt.
Now 1/[P(30 - P)] = 1/30[1/P + 1/(30 - P)] so the DE becomes

[1/P + 1/(30 - P)] dP = 30(0.03) dt and integrating gives
ln|P| - ln|(30 - P)| = 0.9t + c
ln(P/(30 - P)) = 0.9t + c
P/(30 - P) = Ae^(0.9t) where A = e^c
P = Ae^(0.9t)*(30 - P)
P[1 + Ae^(0.9t)] = 30Ae^(0.9t)
so the general solution is
P(t) = 30Ae^(0.9t)/[1 + Ae^(0.9t)].

(b)
When t = 0, P = 5 so
5 = 30A/[1 + A]
5[1 + A] = 30A
25A = 5
so A = 0.2 and the particular solution is
P(t) = 6e^(0.9t)/[1 + 0.2e^(0.9t)].

(c)
The number eventually having the disease will be the limit of P as t--> infinity.
Now P(t) = 6e^(0.9t)/[1 + 0.2e^(0.9t)]
= 6/[e^(-0.9t) + 0.2]
--> 6/0.2 = 30 as t--> infinity since e^(-0.9t)-->0 as t-->infinity.
So 30 000 people will eventually have the disease.I've tried to do the work, please just let me know if i have anything wrong. Thanks guys.

Looks perfectly fine to me.

Bumpp - anyone else!?

What is calculus?

Calculus is a branch of mathematics that deals with the study of change and motion. It involves the use of mathematical concepts such as limits, derivatives, and integrals to solve problems related to rates of change and optimization.

Why is mental creativity and flexibility important in solving calculus problems?

Calculus problems often involve complex and abstract concepts that require out-of-the-box thinking and problem-solving skills. Mental creativity and flexibility are essential in finding new and efficient ways to approach these problems and come up with accurate solutions.

What are some strategies for improving mental creativity and flexibility in solving calculus problems?

Some strategies for improving mental creativity and flexibility include practicing regularly, breaking down problems into smaller parts, and looking for alternative approaches to solving problems. It is also helpful to constantly challenge yourself with new and more difficult problems.

How can I apply calculus in real-life situations?

Calculus has many practical applications in various fields such as physics, engineering, economics, and statistics. It can be used to analyze and optimize systems, predict outcomes, and solve real-world problems involving rates of change, optimization, and motion.

What are some common mistakes to avoid when solving calculus problems?

Some common mistakes to avoid when solving calculus problems include not understanding the problem, not paying attention to details, and not checking your work. It is also important to be familiar with the rules and formulas of calculus and to avoid making arithmetic errors.

Replies
3
Views
790
Replies
7
Views
5K
Replies
1
Views
1K
Replies
7
Views
1K
Replies
12
Views
1K
Replies
9
Views
2K
Replies
4
Views
2K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
14
Views
2K