WEBVTT
00:00:00.840 --> 00:00:03.360
Yvette wants to put some books on a bookshelf.
00:00:04.200 --> 00:00:11.800
The bookshelf has a length of 43 centimetres, a width of 16 centimetres, and a height of 27 centimetres.
00:00:12.080 --> 00:00:14.480
These dimensions are to the nearest centimetre.
00:00:15.600 --> 00:00:24.400
The books all have a length of 2.6 centimetres, a width of 15.3 centimetres, and a height of 25.9 centimetres.
00:00:25.160 --> 00:00:27.600
These dimensions are all to the nearest millimetre.
00:00:28.760 --> 00:00:35.080
By continuing the pattern that’s shown, prove that Yvette can fit at least 16 books on the bookshelf.
00:00:36.200 --> 00:00:43.040
So here we also have a diagram showing the bookshelf and some books and also showing how the books are put in the bookshelf.
00:00:43.920 --> 00:00:46.320
Now, at first, this looks like a very simple problem.
00:00:47.720 --> 00:00:56.560
The height of the bookshelf, 27 centimetres, is greater than the height of all the books, 25.9 centimetres, and so it will fit in height-wise.
00:00:57.600 --> 00:01:06.200
Also, this depth or, as it is called in the question, width of 16 centimetres is greater than the 15.3 centimetres’ width of the book.
00:01:06.480 --> 00:01:07.920
And so they’ll fit in that way.
00:01:08.920 --> 00:01:13.160
The only question is how many of these books can be fit in side by side in the shelf.
00:01:13.600 --> 00:01:22.480
And we can get that surely just by dividing 43 centimetres, the length of the bookshelf, by 2.6 centimetres, the length of the book.
00:01:23.560 --> 00:01:25.880
But all the dimensions quoted aren’t exact.
00:01:26.080 --> 00:01:37.480
The dimensions of the bookshelf are given only to the nearest centimetre, whereas the dimensions of the books slightly more accurately are given to the nearest millimetre.
00:01:37.480 --> 00:01:41.360
As a result, the length of the bookcase isn’t 43 centimetres exactly.
00:01:41.760 --> 00:01:45.200
It’s 43 centimetres to the nearest centimetre.
00:01:46.200 --> 00:01:49.200
And the same is true for the length or thickness of the book.
00:01:50.200 --> 00:01:54.160
This is 2.6 centimetres to the nearest millimetre.
00:01:55.280 --> 00:02:06.960
And so if the bookshelf is slightly shorter than the 43 centimetres and the books are slightly thicker than 2.6 centimetres, we might not be able to fit 43 divided by 2.6 books in the bookshelf.
00:02:07.880 --> 00:02:13.920
Now we can’t say what the values of these dimensions are exactly, but we can use inequalities to write a range of values.
00:02:14.800 --> 00:02:18.760
The length of the bookshelf is 43 centimetres to the nearest centimetre.
00:02:19.600 --> 00:02:23.960
So when we round this length to the nearest centimetre, we get 43 centimetres.
00:02:24.720 --> 00:02:32.600
This length must therefore be less than 43.5 centimetres and greater than or equal to 42.5 centimetres.
00:02:33.560 --> 00:02:37.600
All lengths in this range are 43 centimetres to the nearest centimetre.
00:02:38.440 --> 00:02:39.840
How about the thickness of the book?
00:02:39.840 --> 00:02:43.880
Well, we’re told that it’s 2.6 centimetres to the nearest millimetre.
00:02:44.920 --> 00:02:45.960
We have to be careful here.
00:02:46.360 --> 00:02:50.040
We’re given the length in centimetres, but the accuracy in millimetres.
00:02:51.280 --> 00:02:59.640
There are 10 millimetres in a centimetre, and so 2.6 centimetres is 2.6 times 10, which is 26 millimetres.
00:03:00.760 --> 00:03:04.360
So the thickness of the book is 26 millimetres to the nearest millimetre.
00:03:05.240 --> 00:03:11.280
So it is greater than or equal to 25.5 millimetres and less than 26.5 millimetres.
00:03:11.920 --> 00:03:15.400
We can convert these values back to centimetres by dividing by 10.
00:03:16.200 --> 00:03:25.600
25.5 millimetres is 2.55 centimetres, and 26.5 millimetres is 2.65 centimetres.
00:03:26.880 --> 00:03:31.960
Okay, now remember that we wanted to prove that Yvette can fit at least 16 books on the bookshelf.
00:03:32.840 --> 00:03:38.040
In the worst-case scenario, the thickness of one book is 2.65 centimetres.
00:03:38.760 --> 00:03:40.960
That’s the maximum thickness of a book.
00:03:41.800 --> 00:03:50.760
So the maximum thickness of 16 books or the maximum length if the books are placed side by side as in the diagram is 16 times this.
00:03:51.520 --> 00:03:56.000
And using a calculator, we find this to be 42.4 centimetres.
00:03:56.600 --> 00:04:03.200
And this is less than 42.5 centimetres, which is the minimum possible length of the bookshelf.
00:04:03.880 --> 00:04:15.400
So even if the books are as thick as they’re allowed to be and the bookshelf is as short as it’s allowed to be, you can still fit 16 books inside the bookshelf in the way shown in the diagram.
00:04:16.520 --> 00:04:20.400
We should probably also check that the books really do fit in the other dimensions too.
00:04:21.320 --> 00:04:47.280
It’s a very similar process, slightly easy in fact, to show that the maximum height of a book is 25.95 centimetres, which is less than the 26.5-centimetre minimum height of the shelf, and so the book fits in height-wise, and also that the minimum width of a book is 15.35 centimetres, which is less than the 15.5-centimetre minimum width of the shelf.
00:04:48.160 --> 00:04:55.640
We probably want to just state that we’ve proved what we set out to prove that Yvette can fit at least 16 books on the bookshelf.
00:04:56.800 --> 00:05:10.760
The key to this question was to use inequalities to find the maximum dimensions of the 16 books placed in a row as shown in the diagram and to show that each of these dimensions was less than the corresponding minimum dimension for the bookshelf.