Air flows into a balloon at a rate of r(t) = 2t + 10 litres per… Step by Step

Step by step

The question

Question: Air flows into a balloon at r(t) = 2t + 10 litres per second.

How much flows in from t = 2 to t = 6?

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 2 to 6.

= ∫ from 2 to 6 of (2t + 10) dt

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

Now we have t² + 10t.

= (t² + 10t) from 2 to 6

Put in 6, then take away the same with 2.

= (6² + 10(6)) − (2² + 10(2))

That is 96 minus 24: 72.

= 96 − 24 = 72

So 72 litres flow in.

amount = 72 litres

Careful!

r is the rate, not the amount.

The amount is the integral of the rate.

Careful: not r(6) − r(2) ✗

2 Check

Let's check with a calculator's integral: about 72.

Right!

calculator: ∫ from 2 to 6 of r(t) dt ≈ 72

3 Pick the choice

Our answer is 72 litres.

That is choice C.

C) 72

4 Why not the others

A is 22.

Wrong: that is r(6), the rate at t = 6.

A) 22 ✗ that is r(6), the rate at t = 6

B is 40.

Wrong: lost the t² part.

B) 40 ✗ lost the t^2 part

D is 76.

Wrong: put in t = 2 for 10t only.

D) 76 ✗ put in t = 2 for 10t only

Let's look back.

First: the net change.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer