Differential Equation problem from Key Curriculum Press

[banglajihwan]

Homework Statement

This problem is from Key Curriculum Press Calculus chapter 7, R3 e.

Question: Memory Retention Problem: Paula Lopez starts her campaign for election to state senate. She meets people at a rate of about 100 per day, and she tries to remember as many names as possibl. She finds taht after seven full days, she remembers the names of 600 of the 700 people she met.

i. assume that the rate of change in the number of names she remembers, dN/dt, equals 100 minus an amount that is directly proportional to N. Wrtie a differential equation that expesses this assumption, and solve the equation subject to the initial condition that she knew no names when t=0

I am not sure how to calulate the k value.

Homework Equations

solving differential equations using separation of variables and intergration

The Attempt at a Solution

using the method metioned above, I managed to get:
100/k (1-e^(-kt)) =N
because c= 100 --> using initial conditions
i got 0.045236 for k using the calculator (finding intersection point)
Is there any other way to calculate k without using the calculator?

 

[mighty2000]

Thank you for sharing this interesting problem. I can provide some insight and guidance on how to approach this problem.

First, let's start by writing the differential equation based on the given information. We know that the rate of change in the number of names Paula remembers, dN/dt, is equal to 100 minus an amount directly proportional to N. This can be expressed as:

dN/dt = 100 - kN

where k is the proportionality constant. Now, to solve this differential equation, we can use the method of separation of variables. This involves separating the variables on either side of the equation and integrating both sides. This can be written as:

1/(100 - kN) dN = dt

Integrating both sides, we get:

ln(100 - kN) = t + C

where C is the constant of integration. Now, using the initial condition that Paula knew no names when t=0, we can substitute N=0 and t=0 into the above equation to solve for C. This gives us:

ln(100) = 0 + C
C = ln(100)

Substituting this back into our previous equation, we get:

ln(100 - kN) = t + ln(100)

Now, we can solve for k by taking the exponential of both sides:

100 - kN = e^(t+ln(100))

Simplifying this, we get:

100 - kN = 100e^t

Solving for k, we get:

k = (100 - 100e^t)/N

Finally, using the given information that after 7 days, Paula remembers 600 out of 700 names, we can substitute N=600 and t=7 into the above equation to solve for k. This gives us:

k = (100 - 100e^7)/600

Using this method, we can calculate the value of k without using a calculator. I hope this helps you in solving the problem. Good luck!