289. MAGNITUDES WHICH ARE PROPORTIONAL TO THE SQUARES OF OTHER MAGNITUDES.

The areas of two squares are to each other as the squares of their lengths.

The areas of two circles are to each other as the squares of their diameters.

Observe that the ratio of the areas of the above squares is (or ). But the area of each circle is about (more accurately, .785+) of its circumscribed square; so the ratio of the areas of the circles is (or §).

1. The area of a 6-inch circle is how many times as great as the area of a 3-inch circle ?

2. If a 4-inch circle of brass plate weighs 3 ounces, how much will a 6-inch circle weigh, the thickness being the same in each case?

3. If a piece of rolled dough 1 foot in diameter is enough for 17 cookies, how many cookies can be made from a piece 2 feet in diameter, the thickness of the dough and the size of the cookies being the same in each case?

4. If a piece of wire of an inch in diameter will sustain a weight of 1000 lbs., how many pounds will a wire of an inch in diameter sustain ?

Proportion.

290. MAGNITUDES WHICH ARE PROPORTIONAL TO THE CUBES OF OTHER MAGNITUDES.

The solid contents of two cubes are to each other as

the cubes of their lengths.

The solid contents of two spheres are to each other

as the cubes of their diameters.

Observe that the ratio of the solid contents of the above cubes is

(or 27). But the solid content of each sphere is about (more accurately, .5236-) of its circumscribed cube; so the ratio of the solid contents of the spheres is (or 27).

1. The solid content of a 6-inch sphere is how many times as great as the solid content of a 3-inch sphere ?

2. If a 4-inch sphere of brass weighs 10 lbs., how many pounds will a 6-inch sphere of brass weigh?

3. If a sphere of dough 1 foot in diameter is enough for 20 loaves of bread, how many loaves can be made from a sphere of dough 2 feet in diameter ?

4. If the half of a solid 8-inch globe weighs 4 lbs., how much will the half of a solid 5-inch globe weigh, the material being of the same quality?

291. MAGNITUDES WHICH ARE INVERSELY PROPORTIONAL TO OTHER MAGNITUDES OR TO THE SQUARES OF OTHER MAGNITUDES.

EXAMPLE.

If 5 men do a piece of work in 16 days, how long will it take 8 men to do a similar piece of work?

Operation and Explanation.

It is evident that the time required will be inversely proportional to the number of men employed; that is, if twice as many men are employed, not twice as much, but as much time will be required. Hence the proportion is not 5:8 16: x, but, 5:8x: 16; hence, 5:8: = 10:16.

=

The interpretation of the above equation is, if 5 men can do a piece of work in 16 days, 8 men can do it in 10 days.

1. If 4 men can do a piece of work in 20 days, how long will it take 5 men to do a similar piece of work?

2. If 8 men can do a piece of work in 12 days, how long will it take 3 men to do a similar piece of work?

It can be shown that the intensity of light upon an object diminishes as the square of the distance between the luminous body and the illuminated object increases; that is, if the distance be twice as great in one case as in another, the intensity is not twice as great, not as great, but as great; if the distances are as 2 to 3 the intensities are, not as 2 to 3, not as 3 to 2, but as 9 to 4. The intensity at 2 feet is as great as at 3 feet.

3. Object A is 15 feet from an incandescent electric light. Object B is 20 feet from the same light. Object C is 30 feet from the same light. (a) How does the

light at B compare with the intensity at A?

intensity of the

(b) How does

the intensity at C compare with the intensity at A?

Algebra.

292. TO FIND THE MISSING TERM OF A PROPORTION WITHOUT FINDING THE RATIO.

The first and fourth terms of a proportion are called the extremes, and the second and third terms, the means; thus, in the proportion 12:68:4, 12 and 4 are the extremes and 6 and 8 are the means.

Observe that in the following proportions the product of the means equals the product of the extremes:

6:38:4; then 6 x 4 = 3 x 8

But a and d are the extremes and b and c the means; hence, in any proportion in which abstract numbers are ememployed, the product of the means equals the product of the

11. If 8 acres of land cost $360, how much will 15 acres cost at the same rate?

12. If 12 horses consume 3500 lb. of hay per month, how many pounds will 15 horses consume?

13. If 11 cows cost $280.50, how many cows can be bought for $433.50 at the same rate?

* Observe that in the solution of concrete problems by the method here given the numbers must be regarded as abstract. It would be absurd to talk or think of finding the product of 15 acres and 360 dollars and dividing this by 8 acres. It is true, however, that the ratio of 8 acres to 15 acres equals the ratio of 360 dollars to dollars. It is also true that in the proportion 8:15 = 360: x, the product of the means is equal to the product of the extremes.