dT/dt = -k(T - 25), T(0) = 95, T(3) = 75, when is T = 35

Step by step

The question

Question: find the time t when T = 35, if dT/dt = −k(T − 25) and T(0) = 95, T(3) = 75.

1 Read the equation

T is the temperature, t is the time, and 25 is the room temperature.

The bigger the gap T − 25, the faster it cools.

2 Separate and integrate

This is a separable equation.

Divide by T − 25, 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 25 to both sides.

3 Use the starting value

Put in t = 0 and T = 95.

e⁰ = 1.

So A = 70: the gap at the start.

The formula: T = 25 + 70e^(−kt).

Careful!

Don't forget the room temperature.

Only the gap, 70, shrinks; T gets close to 25, not to 0.

4 Find k

Now the second value: t = 3, T = 75.

Take away 25 from both sides.

Divide both sides by 70.

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

Divide by −3: k is about 0.1122.

So T ≈ 25 + 70e^(−0.1122t).

5 Answer the question

Put in T = 35.

Now t is the unknown.

Take away 25 from both sides.

Divide both sides by 70.

Take ln of both sides.

Divide.

So t ≈ 17.35.

6 Check

Let's check.

At t = 0: 95, the start.

Right!

Put t ≈ 17.35 back in: about 35.

Right!

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.