Video: Measurement Uncertainties

The sides of a rectangular tile are measured to the nearest centimeter, and they are found to be 6 cm and 8 cm. Rounding to the same number of significant figures that the side lengths were measured to, what is the area of the tile?

02:37

Video Transcript

The sides of a rectangular tile are measured to the nearest centimeter, and they are found to be six centimeters and eight centimeters. Rounding to the same number of significant figures that the side lengths were measured to, what is the area of the tile?

Okay, so in this question, we’ve been told that we’ve got a rectangular tile and the side lengths of this rectangular tile were measured to the nearest centimeter. We’ve been told that these measured values were six centimeters for one of the lengths and eight centimeters for the other length. And we’ve been asked to find the area of the tile rounded to the same number of significant figures that the lengths were measured to.

In other words, even though we’ve been told the lengths were measured to the nearest centimeter, we do not want to round the area of a tile to the nearest centimeter. We want to round it to the same number of significant figures as the length measurements, six centimeters and eight centimeters, and that’s an important point to remember.

But first, let’s recall that the area of a rectangle, we’ll call it 𝐴 subscript a rectangle, is equal to the length of the rectangle multiplied by the width of the rectangle. And we can see in our diagram that this is the length and then this must be the width. Therefore, to find the area of our rectangular tile, we can say that this is equal to six centimeters, which is the length, multiplied by eight centimeters, which is the width.

And then, to evaluate this further, we firstly multiply the numbers, six times eight, and then we multiply the units together, centimeters times centimeters, which gives us square centimeters. Now, because six times eight is 48, we find that the area of our rectangular tile is 48 squared centimeters or 48 centimeters squared.

But remember, this is not our final answer. We need to round our answer for the area to the same number of significant figures that the side lengths were measured to. And we can see that each side length, for example, eight centimeters, is measured to one significant figure because, in this particular case, eight is a significant figure and it’s the only one. And the same is true for six centimeters. It’s also measured to one significant figure. Which means that we need to give our answer for the area to one significant figure as well.

Since this four here, this first number in our value, is a significant figure, this means we’re going to round at this position here. But in order to work out what happens to this four, we need to look at the next number. This value is an eight, and eight is larger than or equal to five, and so our four is going to round up.

In other words then, to one significant figure, the area is 50 square centimeters. And hence, we found the answer to our question. If the side lengths of a rectangular tile are found to be six centimeters and eight centimeters, then the area of this tile, to one significant figure, is 50 square centimeters.