dA/dt = 0.1A, A(0) = 1000, find A(3) Step by Step

Step by step

The question

Question: find A(3), if dA/dt = 0.1A and A(0) = 1000.

dAdt = 0.1A

A(0) = 1000

1 Read the equation

The rate of change is 0.1 times the amount A.

The more there is, the faster it grows.

k > 0: growth.

the rate = 0.1 × A: the more A, the faster it changes

2 Separate and integrate

This is a separable equation.

Divide by A, times dt.

dAA = 0.1 dt

Integrate both sides, as in the separable lesson.

ln |A| = 0.1t + C

Use e on both sides.

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

A = Ae^(0.1t)

3 Use the starting value

Put in t = 0 and A = 1000.

e⁰ = 1.

1000 = Ae^(0)

So A = 1000: the amount at the start.

A = 1000

The formula: A = 1000e^(0.1t).

A = 1000e^(0.1t)

Careful!

This is not a steady rate.

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

Careful: not A = 1000 + 0.1 × 1000t ✗

4 Answer the question

Put in t = 3.

A(3) = 1000e^(0.1 × 3)

The power: 0.3.

= 1000e^(0.3)

So A(3) ≈ 1349.86.

≈ 1349.86

5 Check

Let's check.

At t = 0: 1000, the start.

Right!

A(0) = 1000 × 1 = 1000

And the derivative of the formula is 0.1 times A: the equation again.

Right!

A′ = 1000 × 0.1e^(0.1t) = 0.1 × A

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: answer the question.

Last: check.

Answer