A colony of 50 ants triples every 2 weeks. Write the model. Step by Step

Step by step

The question

Question: A colony of 50 ants triples every 2 weeks.

Write the model.

1 Find the start

t is the time in weeks.

We call the number N.

N = number, t = weeks

At the start no time has passed: t = 0.

Then the number is 50.

That is the start.

start, t = 0: N = 50

2 Find the factor

To triple means three times as much: times 3.

triple = × 3

So the factor is 3: every 2 weeks, we multiply by 3.

factor: × 3 every 2 weeks

3 Make a small table

A small table: t = 0, 2, 4: one step is 2 weeks.

Each step, we multiply by 3: 50 times 3 is 150.

Times 3 again is 450.

After two steps, 4 weeks, we multiplied by 3 two times.

Two times is the power 2: 3 squared.

t = 4: 50 × 3 × 3 = 50 × 3²

4 Count the steps

One step takes 2 weeks.

At t = 2: 2/2 is 1 step.

t = 2: 22 = 1 step

At t = 4: 4/2 is 2 steps.

Like the table.

t = 4: 42 = 2 steps

So in t weeks there are t/2 steps.

We divide the time by 2, the length of one step.

That is the power.

t: t2 steps → × 3^t2

5 Write the model

Start times the factor, to the power t/2.

So the model is N = 50 times 3 to the power t/2.

N = 50 × 3^t2

Careful!

Not 3 to the power 2 t.

That would be 2 steps every week.

One step takes 2 weeks, so we divide: t/2.

Careful: not N = 50 × 3^2t ✗

6 Check

Let's check one more step, t = 6.

6/2 is 3 steps.

3 to the power 3 is 27.

50 times 27 is 1350.

model, t = 6: 50 × 3⁶² = 50 × 3³ = 50 × 27 = 1350

One more step of the table: 450 times 3 is 1350 too.

It works!

table, one more step: 450 × 3 = 1350

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: count the steps.

Then: write the model.

Last: check.

Answer