dT/dt = -k(T - 70), T(0) = 350, T(10) = 250, find k Step by Step

Step by step

The question

Question: find k, if dT/dt = −k(T − 70) and T(0) = 350, T(10) = 250.

dTdt = −k(T − 70)

T(0) = 350, T(10) = 250

1 Read the equation

T is the temperature, t is the time, and 70 is the room temperature.

The bigger the gap T − 70, the faster it cools.

the rate = −k × (T − 70): T − 70 is the gap to the room

2 Separate and integrate

This is a separable equation.

Divide by T − 70, times dt.

dTT − 70 = −k dt

Integrate both sides, as in the separable lesson.

ln |T − 70| = −kt + C

Use e on both sides.

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

T − 70 = Ae^(−kt)

Add 70 to both sides.

T = 70 + Ae^(−kt)

3 Use the starting value

Put in t = 0 and T = 350.

e⁰ = 1.

350 = 70 + Ae^(0)

So A = 280: the gap at the start.

A = 350 − 70 = 280

The formula: T = 70 + 280e^(−kt).

T = 70 + 280e^(−kt)

Careful!

Don't forget the room temperature.

Only the gap, 280, shrinks; T gets close to 70, not to 0.

Careful: not T = 350e^(−kt) ✗

4 Find k

Now the second value: t = 10, T = 250.

250 = 70 + 280e^(−10k)

Take away 70 from both sides.

250 − 70 = 280e^(−10k)

Divide both sides by 280.

914 = e^(−10k)

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

ln(914) = −10k

Divide by −10: k is about 0.0442.

k = ln(914)−10 ≈ 0.0442

5 Check

Let's check.

At t = 0: 350, the start.

Right!

T(0) = 70 + 280 × 1 = 350

With this k, at t = 10 the formula gives 250: the second value.

Right!

T(10) ≈ 70 + 280e^(−0.0442 × 10) ≈ 250

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: find k.

Last: check.

Answer