dT/dt = -0.1(T - 22), T(0) = 72, find T(15) Step by Step

Step by step

The question

Question: find T(15), if dT/dt = −0.1(T − 22) and T(0) = 72.

dTdt = −0.1(T − 22)

T(0) = 72

1 Read the equation

T is the temperature, t is the time, and 22 is the room temperature.

The bigger the gap T − 22, the faster it cools.

the rate = −0.1 × (T − 22): T − 22 is the gap to the room

2 Separate and integrate

This is a separable equation.

Divide by T − 22, times dt.

dTT − 22 = −0.1 dt

Integrate both sides, as in the separable lesson.

ln |T − 22| = −0.1t + C

Use e on both sides.

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

T − 22 = Ae^(−0.1t)

Add 22 to both sides.

T = 22 + Ae^(−0.1t)

3 Use the starting value

Put in t = 0 and T = 72.

e⁰ = 1.

72 = 22 + Ae^(0)

So A = 50: the gap at the start.

A = 72 − 22 = 50

The formula: T = 22 + 50e^(−0.1t).

T = 22 + 50e^(−0.1t)

Careful!

Don't forget the room temperature.

Only the gap, 50, shrinks; T gets close to 22, not to 0.

Careful: not T = 72e^(−0.1t) ✗

4 Answer the question

Put in t = 15.

T(15) = 22 + 50e^(−0.1 × 15)

The power: −1.5.

= 22 + 50e^(−1.5)

A calculator: 50 × e^(−1.5) ≈ 11.16.

≈ 22 + 11.16

So T(15) ≈ 33.16.

≈ 33.16

5 Check

Let's check.

At t = 0: 72, the start.

Right!

T(0) = 22 + 50 × 1 = 72

And the derivative of the formula is −0.1 times the gap T − 22: the equation again.

Right!

T′ = 50 × −0.1e^(−0.1t) = −0.1 × (T − 22)

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: answer the question.

Last: check.

Answer