dm/dt = -km, m(0) = 20, m(4) = 15, find m(10) Step by Step

Step by step

The question

Question: find m(10), if dm/dt = −km and m(0) = 20, m(4) = 15.

dmdt = −km

m(0) = 20, m(4) = 15

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 = 20.

e⁰ = 1.

20 = Ae^(0)

So A = 20: the amount at the start.

A = 20

The formula: m = 20e^(−kt).

m = 20e^(−kt)

4 Find k

Now the second value: t = 4, m = 15.

15 = 20e^(−4k)

Divide both sides by 20.

34 = e^(−4k)

Careful!

k is inside a power.

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

Careful: not k = 1520 × 4 ✗

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

ln(34) = −4k

Divide by −4: k is about 0.0719.

k = ln(34)−4 ≈ 0.0719

So m ≈ 20e^(−0.0719t).

m ≈ 20e^(−0.0719t)

5 Answer the question

Put in t = 10.

m(10) = 20e^(−0.0719 × 10)

The power: −0.7192.

≈ 20e^(−0.7192)

So m(10) ≈ 9.74.

≈ 9.74

6 Check

Let's check.

At t = 0: 20, the start.

Right!

m(0) = 20 × 1 = 20

With this k, at t = 4 the formula gives 15: the second value.

Right!

m(4) ≈ 20e^(−0.0719 × 4) ≈ 15

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