dm/dt = -km, m(0) = 80, half-life 5, find m(12) Step by Step

Step by step

The question

Question: find m(12), if dm/dt = −km and m(0) = 80, with a half-life of 5.

dmdt = −km

Half-life 5 means: after 5, half is left.

So m(5) = 40.

m(0) = 80, half-life 5: m(5) = 40

1 Read the equation

The rate of change is a number k times the amount m.

The more there is, the faster it changes.

the rate = −k × m: the more m, the faster it changes

2 Separate and integrate

This is a separable equation.

Divide by m, times dt.

dmm = −k dt

Integrate both sides, as in the separable lesson.

ln |m| = −kt + C

Use e on both sides.

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

m = Ae^(−kt)

3 Use the starting value

Put in t = 0 and m = 80.

e⁰ = 1.

80 = Ae^(0)

So A = 80: the amount at the start.

A = 80

The formula: m = 80e^(−kt).

m = 80e^(−kt)

4 Find k

Now the second value: t = 5, m = 40.

40 = 80e^(−5k)

Divide both sides by 80.

12 = e^(−5k)

Careful!

k is inside a power.

To bring it down, take ln of both sides first.

Careful: not k = 4080 × 5 ✗

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

ln(12) = −5k

Divide by −5: k is about 0.1386.

k = ln(12)−5 ≈ 0.1386

So m ≈ 80e^(−0.1386t).

m ≈ 80e^(−0.1386t)

5 Answer the question

Put in t = 12.

m(12) = 80e^(−0.1386 × 12)

The power: −1.6636.

≈ 80e^(−1.6636)

So m(12) ≈ 15.16.

≈ 15.16

6 Check

Let's check.

At t = 0: 80, the start.

Right!

m(0) = 80 × 1 = 80

With this k, at t = 5 the formula gives 40: the second value.

Right!

m(5) ≈ 80e^(−0.1386 × 5) ≈ 40

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.

Answer