### Video Transcript

The lengths of the sides of a sheet of paper are measured to be 7.8 centimetres and 14 centimetres. Rounding to an appropriate number of significant figures, what is the area of the sheet? Is it 1) 22 centimetres squared, 2) 110 centimetres squared, 3) 109 centimetres squared, 4) 1.8 centimetres squared, or 5) 6.2 centimetres squared?

So in this question, we’ve got the lengths of the sides of a sheet of paper. And we know that these two lengths are measured to be 7.8 centimetres and 14 centimetres. We are told to round to an appropriate number of significant figures and find the-the area of the sheet. So here is our sheet of paper and here are its dimensions, 7.8 centimetres and 14 centimetres. We’re asked to find the area of the sheet.

To find out the sheet’s area, we know that we have to multiply the length of the sheet by the width of the sheet, which, in other words, happens to be 7.8 centimetres times 14 centimetres. Plugging this into our calculator gives us an answer of 109.2 centimetres squared. But this isn’t our final answer, because we haven’t yet rounded to an appropriate number of significant figures.

Also, as a quick aside, we know that the units are centimetres squared because the two lengths that we had were both in centimetres. So we had 7.8 centimetres times 14 centimetres. And we multiplied the two numbers together to give us 109.2 and the centimetres by the centimetres to give us centimetres squared.

Anyway, let’s get back to all this rounding business. Now, in the question, we’ve been given two quantities, 7.8 centimetres and 14 centimetres. If we look carefully, both of these values have been given to two significant figures. Look, here’s the first significant figure and here’s the second. Similarly, for 14 centimetres, here’s the first significant figure and here’s the second. The rule for physics calculations is that we have to give our final answer to the same number of significant figures as the quantity with the lowest number of significant figures in the question.

Now that sounds a bit confusing. So let’s break that down. Basically, the question will give us lots of different values, usually. We need to find the one value which has the lowest number of significant figures. And we need to give our final answer to that number of significant figures as well. Now in this case, they both have two significant figures. So we don’t need to worry so much. All we need to do is to give our answer to two significant figures.

So in our answer for the area, here’s the first significant figure and here’s the second. Now, it’s the one afterwards, it’s the third significant figure that will tell us what happens to the second one. If this third significant figure is greater than or equal to five, then our second significant figure rounds up. If, however, it’s less than five, then our second significant figure stays the same. So we look carefully at our third significant figure. It’s a nine. Nine is larger than five. So the second significant figure rounds up. The zero is gonna become a one. And so, our answer becomes 110 centimetres squared.

Hence, our final answer is that the area of the sheet is 110 centimetres squared, to two significant figures. That’s option two.