dT/dt = -0.1(T - 4), T(0) = 20, find T(10) Step by Step

Step by step

The question

Question: find T(10), if dT/dt = −0.1(T − 4) and T(0) = 20.

dTdt = −0.1(T − 4)

T(0) = 20

1 Read the equation

T is the temperature, t is the time, and 4 is the room temperature.

The bigger the gap T − 4, the faster it cools.

the rate = −0.1 × (T − 4): T − 4 is the gap to the room

2 Separate and integrate

This is a separable equation.

Divide by T − 4, times dt.

dTT − 4 = −0.1 dt

Integrate both sides, as in the separable lesson.

ln |T − 4| = −0.1t + C

Use e on both sides.

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

T − 4 = Ae^(−0.1t)

Add 4 to both sides.

T = 4 + Ae^(−0.1t)

3 Use the starting value

Put in t = 0 and T = 20.

e⁰ = 1.

20 = 4 + Ae^(0)

So A = 16: the gap at the start.

A = 20 − 4 = 16

The formula: T = 4 + 16e^(−0.1t).

T = 4 + 16e^(−0.1t)

Careful!

Don't forget the room temperature.

Only the gap, 16, shrinks; T gets close to 4, not to 0.

Careful: not T = 20e^(−0.1t) ✗

4 Answer the question

Put in t = 10.

T(10) = 4 + 16e^(−0.1 × 10)

The power: −1.

= 4 + 16e^(−1)

A calculator: 16 × e^(−1) ≈ 5.89.

≈ 4 + 5.89

So T(10) ≈ 9.89.

≈ 9.89

5 Check

Let's check.

At t = 0: 20, the start.

Right!

T(0) = 4 + 16 × 1 = 20

And the derivative of the formula is −0.1 times the gap T − 4: the equation again.

Right!

T′ = 16 × −0.1e^(−0.1t) = −0.1 × (T − 4)

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: answer the question.

Last: check.

Answer