dT/dt = -k(T - 20), T(0) = 90, T(5) = 60, find k Step by Step

Step by step

The question

Question: find k, if dT/dt = −k(T − 20) and T(0) = 90, T(5) = 60.

dTdt = −k(T − 20)

T(0) = 90, T(5) = 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 = 90.

e⁰ = 1.

90 = 20 + Ae^(0)

So A = 70: the gap at the start.

A = 90 − 20 = 70

The formula: T = 20 + 70e^(−kt).

T = 20 + 70e^(−kt)

Careful!

Don't forget the room temperature.

Only the gap, 70, shrinks; T gets close to 20, not to 0.

Careful: not T = 90e^(−kt) ✗

4 Find k

Now the second value: t = 5, T = 60.

60 = 20 + 70e^(−5k)

Take away 20 from both sides.

60 − 20 = 70e^(−5k)

Divide both sides by 70.

47 = e^(−5k)

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

ln(47) = −5k

Divide by −5: k is about 0.1119.

k = ln(47)−5 ≈ 0.1119

5 Check

Let's check.

At t = 0: 90, the start.

Right!

T(0) = 20 + 70 × 1 = 90

With this k, at t = 5 the formula gives 60: the second value.

Right!

T(5) ≈ 20 + 70e^(−0.1119 × 5) ≈ 60

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: find k.

Last: check.

Answer