Step by step
The question
Question: find T(10), if dT/dt = −0.1(T − 20) and T(0) = 90.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Read the equation
T is the temperature, t is the time, and 20 is the room temperature.
The bigger the gap T − 20, the faster it cools.
2 Separate and integrate
This is a separable equation.
Divide by T − 20, times dt.
Integrate both sides, as in the separable lesson.
Use e on both sides.
e^C is just a number: call it A.
Add 20 to both sides.
3 Use the starting value
Put in t = 0 and T = 90.
e⁰ = 1.
To get A alone, we take away 20 on both sides.
So A = 70: the gap at the start.
The formula: T = 20 + 70e^(−0.1t).
Careful!
Don't forget the room temperature.
Only the gap, 70, shrinks; T gets close to 20, not to 0.
The right way: T = 20 + 70e^(−0.1t).
4 Answer the question
Put in t = 10.
The power: 0.1 times 10 is −1.
A calculator: 70 × e⁻¹ ≈ 25.75.
So T(10) ≈ 45.75.
5 Check
Let's check.
At t = 0: 90, the start.
Right!
And the derivative of the formula is −0.1 times the gap T − 20: the equation again.
Right!
6 Pick the choice
Our answer is T(10) ≈ 45.75.
That is choice B.
7 Why not the others
A is 33.11.
Wrong: let T go down to 0: T − 20 decays, then add the 20 back.
That is the Careful mistake from before.
C is 20.
Wrong: that is the room, reached only after a long time.
D is 53.11.
Wrong: used 90 e⁻¹: it is (90 − 20) e⁻¹.
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.