Water flows into a tank at a rate of r(t) = 3t^2 + 2 litres per… Step by Step

Step by step

The question

Question: Water flows into a tank at r(t) = 3t² + 2 litres per minute.

How much flows in from t = 1 to t = 3?

There are four choices.

First we solve it, the normal way.

Then we pick.

1 The net change

r(t) is a rate: how fast it flows in.

The rule: the amount is the integral of the rate.

amount = ∫ from a to b of r(t) dt

Here it goes from 1 to 3.

= ∫ from 1 to 3 of (3t² + 2) dt

The antiderivative: add 1 to each power, divide by the new power.

Now we have t³ + 2t.

= (t³ + 2t) from 1 to 3

Put in 3, then take away the same with 1.

= (3³ + 2(3)) − (1³ + 2(1))

That is 33 minus 3: 30.

= 33 − 3 = 30

So 30 litres flow in.

amount = 30 litres

Careful!

r is the rate, not the amount.

The amount is the integral of the rate.

Careful: not r(3) − r(1) ✗

2 Check

Let's check with a calculator's integral: about 30.

Right!

calculator: ∫ from 1 to 3 of r(t) dt ≈ 30

3 Pick the choice

Our answer is 30 litres.

That is choice D.

D) 30

4 Why not the others

A is 29.

Wrong: that is r(3), the rate at t = 3.

A) 29 ✗ that is r(3), the rate at t = 3

B is 26.

Wrong: lost the + 2, which gives 2t.

B) 26 ✗ lost the + 2, which gives 2t

C is 33.

Wrong: used the top limit only.

C) 33 ✗ used the top limit only

Let's look back.

First: the net change.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer