dm/dt = -km, m(0) = 100, half-life 8, find k Step by Step

Step by step

The question

Question: find k, if dm/dt = −km and m(0) = 100, with a half-life of 8.

dmdt = −km

Half-life 8 means: after 8, half is left.

So m(8) = 50.

m(0) = 100, half-life 8: m(8) = 50

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 = 100.

e⁰ = 1.

100 = Ae^(0)

So A = 100: the amount at the start.

A = 100

The formula: m = 100e^(−kt).

m = 100e^(−kt)

4 Find k

Now the second value: t = 8, m = 50.

50 = 100e^(−8k)

Divide both sides by 100.

12 = e^(−8k)

Careful!

k is inside a power.

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

Careful: not k = 50100 × 8 ✗

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

ln(12) = −8k

Divide by −8: k is about 0.0866.

k = ln(12)−8 ≈ 0.0866

5 Check

Let's check.

At t = 0: 100, the start.

Right!

m(0) = 100 × 1 = 100

With this k, at t = 8 the formula gives 50: the second value.

Right!

m(8) ≈ 100e^(−0.0866 × 8) ≈ 50

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: find k.

Last: check.

Answer