dT/dt = -k(T - 20), T(0) = 100, T(10) = 60, find T(20) Step by Step

Step by step

The question

Question: find T(20), if dT/dt = −k(T − 20) and T(0) = 100, T(10) = 60.

dTdt = −k(T − 20)

T(0) = 100, T(10) = 60

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.

the rate = −k × (T − 20): T − 20 is the gap to the room

2 Separate and integrate

This is a separable equation.

Divide by T − 20, times dt.

dTT − 20 = −k dt

Integrate both sides, as in the separable lesson.

ln |T − 20| = −kt + C

Use e on both sides.

e^C is just a number: call it A.

T − 20 = Ae^(−kt)

Add 20 to both sides.

T = 20 + Ae^(−kt)

3 Use the starting value

Put in t = 0 and T = 100.

e⁰ = 1.

100 = 20 + Ae^(0)

So A = 80: the gap at the start.

A = 100 − 20 = 80

The formula: T = 20 + 80e^(−kt).

T = 20 + 80e^(−kt)

Careful!

Don't forget the room temperature.

Only the gap, 80, shrinks; T gets close to 20, not to 0.

Careful: not T = 100e^(−kt) ✗

4 Find k

Now the second value: t = 10, T = 60.

60 = 20 + 80e^(−10k)

Take away 20 from both sides.

60 − 20 = 80e^(−10k)

Divide both sides by 80.

12 = e^(−10k)

Take ln of both sides: ln and e cancel, so the power comes down.

ln(12) = −10k

Divide by −10: k is about 0.0693.

k = ln(12)−10 ≈ 0.0693

So T ≈ 20 + 80e^(−0.0693t).

T ≈ 20 + 80e^(−0.0693t)

5 Answer the question

Put in t = 20.

T(20) = 20 + 80e^(−0.0693 × 20)

The power: −1.3863.

≈ 20 + 80e^(−1.3863)

A calculator: 80 × e^(−1.3863) ≈ 20.

≈ 20 + 20

So T(20) ≈ 40.

≈ 40

6 Check

Let's check.

At t = 0: 100, the start.

Right!

T(0) = 20 + 80 × 1 = 100

With this k, at t = 10 the formula gives 60: the second value.

Right!

T(10) ≈ 20 + 80e^(−0.0693 × 10) ≈ 60

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: find k.

Then: answer the question.

Last: check.

Answer