The cube has side 10 cm, measured with a possible error of 0.1 cm… Step by Step

Step by step

The question

Question: The cube has side 10 cm, measured with a possible error of 0.1 cm.

Estimate the percentage error in the volume.

1 Draw and name

The side is s = 10 cm.

s = 10 cm

The possible error in it is ds = 0.1 cm: a small change.

ds = 0.1 cm

2 Find dV

The volume of a cube is V = s³.

V = s³

Take the derivative, times ds: dV = 3s² ds.

dV = 3s² ds

The change in the volume is about the derivative times the change in the side.

dV = (derivative) × ds

3 Put the numbers in

Put in s = 10 and ds = 0.1.

dV = 3(10)²(0.1)

So the error in the volume is about 30 cm³.

= 30 cm³

Careful!

A small error in the side makes a bigger error in the volume.

Careful: dV is not ds: the error grows

4 The error

The percentage error compares the error with the whole volume.

percentage error = dVV × 100%

The whole volume is 1000 cm³.

So the percentage error is about 3%.

= 301000 × 100% = 3%

5 Check

Let's check with the real change: the volume at 10.1, minus the volume at 10, is 30.3 cm³.

Close to our 30.

Right!

V(10.1) − V(10) = 30.3 cm³

Let's look back.

First: draw and name.

Then: find dV.

Then: put the numbers in.

Then: the error.

Last: check.

Answer