9. A cliff 40 feet high overhangs a river. A man on the cliff throws a stone vertically upward with a velocity of 30 ft. per second. If the height of the stone above the cliff after t seconds is given by

S= 16t2 + 30t,

find how high the stone rises, and when it will strike the water.

10. If the perimeter of a rectangle is 8 feet, find the area as a function of one of the dimensions. Construct the graph of the area, and find the dimensions and area of the largest rectangle of this sort.

11. Solve Exercise 10 if the perimeter is any constant p. [The parcel post regulations require that the sum of the length and girth (greatest perimeter of a section at right angles to the length) of a package shall not exceed 6 feet. By means of this Exercise, what can be said of the shape of the largest rectanglar box which can be mailed?]

12. The equation of the path of a ball thrown into the air at an angle of 60° with the horizon with a velocity of 32 feet per second is

Construct the graph, find the greatest height attained, and where the ball will hit the ground.

13. If a body is projected vertically upward with an initial velocity of v feet per second, its height s after t seconds is given by the equation - 16t2 + vt.

Find how long the body will rise and its maximum height. Prove that the time of rising equals the time of falling.

14. A farmer estimates that if he digs his potatoes now he will have 100 bushels worth $1.25 a bushel; but that if he waits, the crop will increase 16 bushels a week, while the price will drop 8¢ a week. Find the value of his crop as a function of the time in weeks, and draw the graph. When should he dig to get the greatest cash returns?

15. The amount of wheat obtained per acre depends on the intensity of cultivation. A farmer finds that he can cultivate 15 acres with sufficient intensity so that the return will be 30 bushels per acre; 20 acres so that the return will be 25 bushels per acre; 25 acres so that the return will be 20 bushels per acre; etc. Find the law giving the total return as a function of the number of acres under cultivation, and plot the graph. What is the best size acreage for him to cultivate in order to get the largest gross returns?

35. Empirical Data Problems. The table of values of a linear function is such that if the successive values of Ax are equal so also are the successive values of Ay.

Any quadratic function has a somewhat analogous property, which is illustrated in the table for y 2x2 + 3x + 4. The

Ax X

1

y Ay A2y

0

4

1 9

LO

5

values of Ax being equal, those of Ay, sometimes called the first differences of y, are not equal. But the successive differences of Ay, called the second differences of y and denoted by A2y, are equal. Every quadratic function pos- 1 sesses this property, that if the values of x are such that the successive values of Ax are equal, then the successive values of the second differences oj y, ▲2y, are equal.

4

9

2 18

4

13

3 31

4

1

17

4 48

1

This property enables us to tell whether a given table of values may be represented approximately by a quadratic function, and also to determine the coefficients of the function.

EXAMPLE. Find a quadratic function which represents approximately the law connecting the values of x and y given in the table.

Inspection of the values of x shows that the successive values of Ax are equal. Computing the successive values of Ay and A2y, 7.6 it is found that the latter are nearly equal. Hence the values of y resemble those of a quadratic function.

х

Equating the average values of A2y, 8a= 8.1, whence a = 1.01. Equating the average values of Ay, 24a + 2b = 20.08. Substituting the value of a, and solving for b, we get b 2.08.

Equating the average values of y, 44a + 6b + c = 35.12. Substituting the values of a and b, and solving for c, we get c = 3.16. Substituting the

If it is desired to find the equation of a parabola with vertical axis which passes through, or near, several points whose coördinates are given, the method used in the example may be employed even though the values of Ax, and hence also those of A2y, are not equal. At least three points must be given.

EXERCISES

1. Find the equation of the parabola with vertical axis which passes through the points (1, 0),(3, 10), (5, 28). Construct the graph and check the result.

2. In sinking a deep mine, new material as well as labor must be invested each day, and the required depth cannot be determined in advance with accuracy. A company which has set aside $100,000 for the cost of sinking a shaft finds the total capital invested as the work increases as given in the table. Find Time in months ment as a function of the the law giving the invest

0, 1, 2, 3,

Investment in thousands 4, 12, 26.1, 46.1,

time. If the work continues to progress according to the same law, when must the work be completed if the cost is not to exceed the amount set aside?

3. A contractor agreed to build a breakwater for $125,000. After spending $33,000 as indicated in the table, he threw up the job without receiving any pay, because he estimated that the cost would increase according to the same law, and that it would require six months more to complete the work. How much would he have lost if he had finished the breakwater? Time in months

0,

Investment in thousands! 5,

1, 6.1,

2, 11.2,

3, 20.2,

4

33.1

4. An electric conductor gives out a definite amount of current in every mile of its length. Let x be the distance of any point in miles from the

end of the line remote from the generator, and y the voltage there. Assuming that the law is of the form y ax2 + b, determine the law from the table. What is the voltage at a point 2.5 miles from the end?

C 0, 1, 2, 3, 4 200, 212.5, 250, 312.5, 400

Hint: The value of a may be determined by the general method; after finding a, substitute its value in У ax2+b. Then a value of b may

be determined by substituting any pair of values of x and y. A better value of b is obtained by determining its value for each pair of values of x and y and averaging the results (compare the method of finding b in Section 27 after m is determined).

36. The Function x". Among the functions represented by xn which may be obtained by assigning a numerical value to n may be mentioned the following:

If no numerical value is assigned to n, the symbol " may represent either the totality of all functions obtained by assigning a numerical value to n, or a particular, but unassigned, one of these functions.

The function ", sometimes called the power function, is considered in the following sections.

37. Tables of Squares, Cubes, Square Roots, Cube Roots, and Reciprocals. These tables are extensive tables of values of the function a" for n = 2, 3, 1, 3, and - 1. They are laborsaving devices, for from them we can find, for example, the square of a number without the labor of multiplying it by itself. We shall use Huntington's Four Place Tables, Unabridged Edition.

Tables of squares and cubes. The square of a number n, between 1 and 10, may be found from the table on page 2. In order to economize room, the table is not arranged in two long columns of rows. Instead, the first digits in n are given in the border on the left and the last digit in the border at the

top of the table. To find the square of 1.56, look in the row in which 1.5 stands on the left, under "n," and in the column headed "6." Here we find 2.434, which is the square of 1.56 to four figures. The exact square, obtained by multiplication, is 2.4336, which is nearer 2.434, as given in the table, than to 2.433. The lack of exactness in the tables is usually immaterial, for in most of the applications of mathematics three or four figure accuracy is all that is desired, and in many it is all that can be attained. Thus if 1.56 is the side of a square, obtained by measurement, the area is 1.562 2.43. Only three figures are retained in the area since the product cannot contain more significant figures than the factors (Section 26). For such an example the table of squares is more accurate than

necessary.

To find the squares of numbers greater than 10, or less than 1, we use, respectively, the relations

(10n)2 = 100n2 and

(n/10)2 = n2/100.

Since multiplication by 10 or 100 shifts the decimal point one or two places to the right, respectively, while division shifts it to the left, we have the rule given at the top of the table: "Moving the decimal point one place in n is equivalent to moving it two places in n2."

2

Repeated application of this rule enables us to find the square of any number. Thus to find 2472, we find from the table 2.472, = 6.10, whence, applying the rule twice, 2472 = 61,000. Similarly, 0.2472 0.0610.

The table of cubes, on page 4, is very similar to the table of squares. The rule at the top of the table for shifting the decimal follows from the relations:

(10n)3 = 1000n3 and

(n/10)3 = n3/1000.

Tables of square roots and cube roots. The table of square roots on page 3 is separated into two parts which give the square roots of numbers from 0.1 to 1 and from 1 to 10. The reason for this lies in the rule for shifting the decimal point.