dm/dt = -0.5m, m(0) = 40, find m(2) Step by Step

Step by step

The question

Question: find m(2), if dm/dt = −0.5m and m(0) = 40.

dmdt = −0.5m

m(0) = 40

1 Read the equation

The rate of change is −0.5 times the amount m.

The more there is, the faster it shrinks.

k < 0: decay.

the rate = −0.5 × m: the more m, the faster it changes

2 Separate and integrate

This is a separable equation.

Divide by m, times dt.

dmm = −0.5 dt

Integrate both sides, as in the separable lesson.

ln |m| = −0.5t + C

Use e on both sides.

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

m = Ae^(−0.5t)

3 Use the starting value

Put in t = 0 and m = 40.

e⁰ = 1.

40 = Ae^(0)

So A = 40: the amount at the start.

A = 40

The formula: m = 40e^(−0.5t).

m = 40e^(−0.5t)

Careful!

This is not a steady rate.

The rate grows with m, so the formula has e, not a plus.

Careful: not m = 40 + −0.5 × 40t ✗

4 Answer the question

Put in t = 2.

m(2) = 40e^(−0.5 × 2)

The power: −1.

= 40e^(−1)

So m(2) ≈ 14.72.

≈ 14.72

5 Check

Let's check.

At t = 0: 40, the start.

Right!

m(0) = 40 × 1 = 40

And the derivative of the formula is −0.5 times m: the equation again.

Right!

m′ = 40 × −0.5e^(−0.5t) = −0.5 × m

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: answer the question.

Last: check.

Answer