dA/dt = 0.08A, A(0) = 2000, find A(12) Step by Step

Step by step

The question

Question: find A(12), if dA/dt = 0.08A and A(0) = 2000.

dAdt = 0.08A

A(0) = 2000

1 Read the equation

The rate of change is 0.08 times the amount A.

The more there is, the faster it grows.

k > 0: growth.

the rate = 0.08 × A: the more A, the faster it changes

2 Separate and integrate

This is a separable equation.

Divide by A, times dt.

dAA = 0.08 dt

Integrate both sides, as in the separable lesson.

ln |A| = 0.08t + C

Use e on both sides.

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

A = Ae^(0.08t)

3 Use the starting value

Put in t = 0 and A = 2000.

e⁰ = 1.

2000 = Ae^(0)

So A = 2000: the amount at the start.

A = 2000

The formula: A = 2000e^(0.08t).

A = 2000e^(0.08t)

Careful!

This is not a steady rate.

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

Careful: not A = 2000 + 0.08 × 2000t ✗

4 Answer the question

Put in t = 12.

A(12) = 2000e^(0.08 × 12)

The power: 0.96.

= 2000e^(0.96)

So A(12) ≈ 5223.39.

≈ 5223.39

5 Check

Let's check.

At t = 0: 2000, the start.

Right!

A(0) = 2000 × 1 = 2000

And the derivative of the formula is 0.08 times A: the equation again.

Right!

A′ = 2000 × 0.08e^(0.08t) = 0.08 × 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