Illustrative Example. 422

= ?

(a) 17.

(b) 8.

(c) 64. (d) 1764.

2. To square numbers from 50 to 75.

(a) Subtract 50 from the number.

(b) Add the result to 25.

(c) Square the first result, and

(d) Add it to the second result considered as hundreds.

3. To square numbers from 75 to 100.

(a) Subtract the number from 100.

(b) Subtract the first result from the number.

(c) Square the first result, and

(d) Add it to the second result considered as hundreds.

Illustrative Example. 882=?

(a) 12. (b) 76.

(c) 144.

(d) 7744.

NOTE. Inquisitive students will desire to find the reasons for these rules.

The first is an application of this formula: (a — b)2 = a2 · 2ab+b2. a here represents 50, and b the difference between 50 and the given number.

The second is an application of (a + b)2 = a2 + 2 a b + b2; the letters being used as in the first.

The third employs (a - b)2; a representing 100, and b the difference between 100 and the given number.

1. 86004573 = ? What is the remainder?

2. .0685? Carry root to thousandths.

3. V2=? Carry root to hundredths.

4. V.5=?

6. 15.625 ?

5. 3698.400375 = ?

=

7. √448=?

NOTE. Teachers should dictate many problems until pupils can work rapidly and accurately.

24. A cubical block contains 96 feet of lumber. How many inches long is each edge?

25. A cubical cistern holding 3,600 gallons is how deep? 26. Find the dimensions of a cubical bin whose capacity is 2,000 bushels.

346. 1. The method of extracting the cube root of a number may be illustrated by the use of blocks.

A cubical figure contains 50,653 cubic inches. What is the length of one edge?

3. The number of inches in the edge is a two-place number. The largest cube in 50,000 is 27,000. 27,000 inch-cubes will form a figure each of whose edges is 30 inches. This 27,000 is "the cube of the tens," the first part of the expansion shown in Art. 338, 2. 23,653 inch-cubes remain with which to enlarge the figure. The size must be increased in such a way as to retain the form of a cube. Since the length, breadth, and thickness are equal, the additions must be made to three adjacent faces, and must be equal. If a layer of cubes were placed upon one face, 30 X 30 900 cubes would be required. Since three such additions are to be made, 2,700 cubes would be needed to make the additions one inch thick on each face. We thus find the illustration of "three times the square of the tens as a trial divisor." It is called a "trial divisor" because we wish to ascertain how many such additions may be made to each face with the remaining blocks, and yet leave enough to fill out the figure.

The indications are that 7 such layers may be added to each face. This would use up 3 X 30 X 7 blocks. The expression is "three times the square of the tens by the units," or the second part of the expansion shown in Art 338, 2.

3 X 302 X 7 = 18,900. 23,653 18,900 4,753, the number of inchcubes remaining.

The figure is now 37 inches wide, 37 inches long, and 37 inches high, but is not a cube, since additions are still to be made along three edges and on one corner. Each of these additions to the edges must be 7 inches by 7 inches and 30 inches high. These will require 3 X 30 X 72 = 4,410 blocks. This expression is "three times the tens by the square of the units," or the third part of the expansion shown in Art. 338, 2.

An unfilled corner, 7 inches by 7 inches by 7 inches, remains. Since 343 blocks are needed to make the figure a cube, it is clear that 50,653 is the cube of 37. The 343 is "the cube of the units," or the last part of the expansion shown in Art 338, 2.

EXPLANATION OF THE FIGURES.

Fig. 1 represents the cube formed from 27,000 blocks.

Fig. 2 represents the three additions made to the faces of Fig. 1. Fig. 3 represents the figure resulting from the three additions to the faces of Fig. 1.

Fig. 4 represents the three additions made to the edges, and

Fig. 5 the final addition made to the corner to complete the cube.

Apply this method of illustration to Problems 1-6, Art. 339.

1. What is the area of a piece of ground arranged in the form of a triangle, the length of one side being 124 rods, and the distance to the vertex of the opposite angle being 86 rods?

2. How many feet of lumber are there in a two-inch plank in the form of an isosceles triangle, 18 feet long and 14 inches wide at the base?

3. Find the area in acres of the irregular field A B C D E.

4. A board in the form of a rhombus is 19 inches on each side and 10 inches high. Draw the figure. What is its area? Compare it with a square that is 19 inches on each side.

NOTE. A Rhombus is an oblique-angled equilateral parallelogram.

5. A field in the form of a trapezoid contains 23 acres. One of its parallel sides is 95 rods and the other 65 rods long. What is its width?

NOTE 1. A Trapezoid is a quadrilateral only two of whose sides are parallel. Its area is equal to the prodBuct of its height and half the sum of its parallel sides.

A G

6. What is the area of a circular pond whose diameter is 15 rods?

7. What is the circumference of a circus ring whose area is 872 square yards?

8. What is the area of a field in the form of a trapezoid, the parallel sides being 42 rods and 88 rods respectively, and the distance between these sides being 36 rd. 3 yd.?

9. A cubical box contains 79,507 cubic inches. the length of each edge?

What is

10. A cubical tank has a capacity of 760 gallons. Find the length of one side. (Approximate.)

11. A cubical cistern is 9 feet deep. What will it cost to construct it at 75 cents a barrel (31 gallons)?

12. A corn-bin whose width equals its height is 4 times as long as it is high. If it will hold 3,200 bushels of shelled corn, what is its length?

13. A building is 36 feet wide. If the attic is 9 feet high, what is the length of the rafters, allowing for a projec+ion of 18 inches?

14. If the area of an equilateral triangle is 12 square feet, what is the area of a similar triangle each of whose sides is twice as long? Four times as long? 7 times as long?