Step by step
The question
Question: find P(5), if dP/dt = 0.03P and P(0) = 800.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Read the equation
The rate of change is 0.03 times the amount P.
The more there is, the faster it grows.
k > 0: growth.
2 Separate and integrate
This is a separable equation.
Divide by P, times dt.
Integrate both sides, as in the separable lesson.
Use e on both sides.
e^C is just a number: call it A.
3 Use the starting value
Put in t = 0 and P = 800.
e⁰ = 1.
So A = 800: the amount at the start.
The formula: P = 800e^(0.03t).
Careful!
This is not a steady rate.
The rate grows with P, so the formula has e, not a plus.
The right way: P = 800e^(0.03t).
4 Answer the question
Put in t = 5.
The power: 0.03 times 5 is 0.15.
So P(5) ≈ 929.47.
5 Check
Let's check.
At t = 0: 800, the start.
Right!
And the derivative of the formula is 0.03 times P: the equation again.
Right!
6 Pick the choice
Our answer is P(5) ≈ 929.47.
That is choice D.
7 Why not the others
A is 920.
Wrong: grew by 0.03 × 800 each year in a straight line.
That is the Careful mistake from before.
B is 824.36.
Wrong: used t = 1: it is 5 years.
C is 927.42.
Wrong: used 1.03⁵: e^(0.15) grows a little faster.
Let's look back.
First: read the equation.
Then: separate and integrate.
Then: use the starting value.
Then: answer the question.
Then: check.
Then: pick the choice.
Last: why not the others.