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Population in Millions
19. The population curves, 1, 2 and 3 (in Fig. 12), are those of England, Ireland, and Scotland respectively. What was the population of each of those countries in 1841? When was the population of Ireland decreasing most rapidly, and that of England
increasing most rapidly? Forecast from the curve the population of England in 1911.
(Examples of reading: The population of Scotland in 1831 was 2 millions, that of Ireland in 1821 was 6.8 millions, and that of England in 1871 was 22.5 millions.)
20. The Navy expenditure is given below for certain years. Plot these on a diagram and draw a broken line through the points. Draw also a straight line to lie as near this broken line as possible, and taking this straight line to indicate what the expenditure would be in 1905-6, and in 1906-7, if there was no change of plans, estimate the expenditure for each of these years. If the population may be estimated as 44 millions in 1906–7, what would the cost per head be in that year?
21. One man walks up a street 600 yards long at the rate of 200 yards in three minutes, but stops for five minutes in a shop halfway up the street. Another man, starting from the same end as the first, but seven minutes later, walks after him at the rate of 100 yards per minute. Draw lines on the squared paper which will indicate their respective distances from the end of the street at given times, using 1 inch to represent two minutes, and 1 inch to represent 100 yards.
Hence, or otherwise, find where and when the second man will overtake the first.
1. How shall we measure the air-space of the schoolroom; how much oil an oil-drum holds; how much coal can be mined from a given seam; or how much water a swimmingbath or a reservoir holds? What sort of unit do we need for these?
Let us begin with the schoolroom.
Our unit must be a piece of space, such as a cubic metre, or a cubic foot. These are suitable for measuring the volume of a room. For smaller volumes the litre, gallon, cubic inch, cubic centimetre, &c., are useful.
Make a decimetre cube and an inch cube of paper by cutting out suitable
pieces of paper, folding them, and pasting them together (see Fig. 1); leave margins for the pasting. If you can get pasteboard make a foot cube from it. In a corner of the room mark with chalk how a metre cube would stand.
Measure in metres the length, breadth, and height of the room you
are in, which we will suppose rectangular. Suppose them to be 13 metres, 10 metres, and 4 metres, neglecting fractions for the present. (The teacher will of course use the actual dimensions in place of these.) If you covered
the floor with a layer of metre cubes, how many would there be, and what would be their volume ?—130 cubes, and their volume 130 cubic metres. How many
layers would fill the room, how many cubes are there then, and what is the volume or air-space of the room? 4 layers, 520 cubes, and the air-space is 520 cubic metres, or, in terms of the measured dimensions, 13 x 10 x 4 cubic metres.
The building bricks that children use, or clay or plasticine shaped into cubes, should be used as models.
2. Now measure the room more accurately, and suppose we find the dimensions in metres 13.6, 10.2, and 4.4. What additions must be made to the volume calculated?
With the building bricks a model, which is not to scale, may be made by increasing each dimension by a unit, or a model to scale may be made by using pieces of board along with the bricks. Either model shows that we can fill up the space left by
(a) three slabs measuring (in metres) 13 x 10 x 0.4, 10 x 4 x 0.6, 4 x 13 x 0.2;
(b) three pillars measuring 4x0.6 x 0.2, 13 x 0.2 × 0.4, 10 x 0.4 x 0.6;
(c) a little block measuring 0.6 × 0.2 × 0.4.
What is the volume of a slab, say the horizontal one? How many metre cubes would ten such slabs together make ?They would make 13 x 10 x 4 cubes, so that one 13 x 10 x 4 10
slab has a volume of metres.
or 13 x 10 x 0.4 cubic
What is the volume of a pillar, say the vertical pillar? How many such pillars would you put together to be able to cut them up into metre cubes?
And how many copies of the little block would you put together to be able to cut the figure so made into metre cubes?
We can now calculate the volume of each piece, and we see that to find the total volume we carry out operations that could be used to give the product 13'6 x 10.2 × 4.4, or
610-368. Give separately the volumes of the original piece, the slabs, the pillars, and the little block. They are
and, assuming the dimensions of the room to have been measured to the nearest decimetre, we give as an appropriate result for the air-space 610 cubic metres. What is the order of importance of the various volumes you have given? And how many of them affect the result if it is wanted to one significant figure? If to two significant figures?
If you had begun with the decimetre and cubic decimetre as units, what would you have had for dimensions and volume of the room? And what is this volume expressed in cubic metres?
3. Give a formula for the air-space in terms of the dimensions of the room. The air-space and the dimensions being expressed in corresponding units, we have
volume = length x breadth x height,
or, if we denote volume, length, breadth, and height by v, l, b, h,
Need the length, &c., be whole numbers or involve only decimals of a particular kind ?—No; for decimals to any number of places we can proceed by using successive units, smaller and smaller; or we can begin with a unit small enough for the smallest volume that occurs; and in either case we arrive at the formula
v = lxbxh.
Are vulgar fractions also included? Vulgar fractions could be considered for themselves in the same way as decimals, but any vulgar fraction can be expressed decimally