6. Find the diameter of a wheel, the circumference of which is 18 8496 ft. Ans. 6 ft.

7. Find the radius of a wheel, the circumference of which is 94249 ft. Ans. 15,000 ft.

8. How much pasteboard is marked out by a pair of compasses, the legs of which are 3 in. asunder? Ans. 28 27446 in.

PROBLEM VI. To find the area of the

between two concentric circles.

RULE.-Area Difference of areas of two circles.

=

EXAMPLE. Find the area of a gravel path round a circular garden, the total diameter being 20 ft., and the smaller 15 ft.

Or Area (202-15°) ×·7854 = 137·445.

=

EXERCISES.

1. Find the area of the path, if the larger diameter be 6 feet, and the smaller 4. Ans. 15.708.

2. Find the area when the radii are 8 and 5 feet. Ans. 122 5224.

3. Find the area when the radii are 10 875 and 4.75. Ans. 300 6609.

PROBLEM VII.-To find the third side of a right-angled triangle, any two sides being given.

The square of the hypotenuse

the sum of the

squares of the two sides containing the right angle: any two sides, therefore, being given, we can determine the rest.

Thus, if the hypotenuse be 6, and one side 4,

62=42+x2.36-16=x2 x= √20.

Again, if the sides containing the right angle be given to find the hypotenuse,

22=62+5=36+25=61..x=√61.

PROBLEM VIII.-To find the area of a regular polygon. RULE.-Multiply the perimeter (sum of sides) by perpendicular from centre on one side, and divide by two; or Perim. x Perp.

Area:

2

EXAMPLE.-Find the area of a regular pentagon, of which one side is 5 ft.; and the perpendicular from the centre 3.44.

Area =

Perim. x Perp. 5×5×3.44

2

= 43 sq. feet.

=

=25×1.72

2

EXERCISES.

1. Find the area of a pentagon, of which the side is 2 feet, and the perpendicular 1.72 feet. Ans. 10.75 feet.

2. Find the area of a regular hexagon, of which one side is 10 ft., and the perpendicular 8.5. Ans. 255 feet.

3. Required the area of a hexagon, of which one side is 14 6 feet, and the perpendicular from the centre 12 64 feet. Ans. 553 632 feet.

4. Required the area of a regular heptagon, of which one side is 19.38 in., and the perpendicular from the centre 20 in. Ans. 1356 6 sq. in.

5. Find the area of a regular octagon, of which one of the equal sides is 9.941, and the perpendicular from the centre 12. Ans. 477-168.

MISCELLANEOUS QUESTIONS IN MENSURATION.

1. What is the area of an oblong piece of land 50ft. 6in. long, and 5ft. 5in. wide? Ans. 273 ft. 6 in.

2. A piece of marble is 13 ft. 10 in. long, 2ft. 9 in. wide, and 1.ft. 11⁄2 in. thick; what is its solid content? Ans. 42 ft. 11. 1 in.

3. A field measures 642 links in the base, and 1792 links in altitude; what is its area? Ans. 2 rood 12 poles.

4. In a right-angled triangle, the base is 56 feet, and the perpendicular 33; what is the hypotenuse? Ans. 65 ft.

5. What is the area of a triangle, the three sides of which measure 30, 20, and 40 yards? Ans. 290·5 sq. yds. nearly.

6. A ladder 16 yds. long placed in a street reaches a window 9 yds. above the ground, on one side of the street; if the foot of the ladder be retained in its place, and the ladder be made to rest against the house on the opposite side, it is 12 yds. high at the top: what is the width of the street? Ans. 25 374

yards nearly.

7. Required the area of a six-sided field, ABCDEF, of which EB is a diagonal, and is 30 15 ft.; and the perpendiculars let fall on it from F, A, D, and C, are 8.56ft., 9.26ft., 10 56 ft., and 12.24ft. respectively? Ans. 470 4155 sq. ft.

;

8. The earth's circumference is about 25,000 miles what is the length of a degree? Ans. 69 444 miles. 9. Find the area of a circle, the diameter of which is 7 feet. Ans. 38 ft.

10. What is the diameter of a circle, the area of which is one acre? Ans. 78 yds.

11. A cow is tethered so as to feed over one acre of grass; what is the length of the cord by which she is confined? Ans. 39 yds.

12. A street is 25 374 feet in width; how long a ladder will be required which can be so placed in the street, that when it leans upon the houses on one side its top will be 12 feet above the level of the street, and when it is turned to the houses on the other side, its foot being still in the same place, its top is 9ft. 4in. above the level of the street? Ans. 16 ft. 8 in.

13. A piece of land occupies an area of two hundred and seventy-three square feet six inches and a half; its length is fifty feet six inches: what is its breadth ? 5 ft. 5 in.

ALGEBRA.

In Algebra quantities are represented by letters, the earlier letters of the alphabet a, b, c, d, generally standing for known and the later x, y, z, for unknown quantities. The same signs are employed as in arithmetic.

When a=4, b=8, c=10, d=&

Find the value of

1. 5a+6b+2c+d+4a+7b+3c+5a+6b+2d.

When a 8, b=10, c=12, d=4, e=0.

5. ab+5ac-2ad+7e+6de +5b. Ans. 546.

6.

4ac15bd7bce+6cd-4ab-7ae.

Ans. 952.

7. 7abd+15b-8bd-7de +14ab-8bcd+4ac.

8. 14ad-27bcd-17ae+21bd.

9.

10.

11.

12.

13.

When a=4, b=6, c=8, d=10, e=0.

15a2-12bc-3b2c+5c3d-d2c.

14a2d-7a2b2+7c2d2-8e.

Ans. -266.

Ans.-11672

Ans. 23600.

Ans. 43008.

3a2-7ab2+6a2b2-d2. Ans. 2396.

3abcde+4a2e-3a2ed-7de. Ans. 0.

a2-b2-c2—d2-e2 + ab-bc+de. Ans. -208.

When a=16, b=12, c=8, d=10, e=4, f=0.
Find the value of—

(1.) e÷a, e÷b, 2÷(a+b), d÷(b+c), f÷(a+3). Ans.,,,, 0.

(2.) (bcd+adf)÷ce + (b2-c2 + d2) ÷ d + abc÷ (a+b+c.) Ans. 903.

(3.) (c3-4d) ÷ d + (c―e + d) ÷ (a+b) (a+b+c)÷(a-b-c.) Ans. -101. (a2+b2)÷(2cd+ƒ)+(c2+e2)÷2ab + (3b3—c) 4. Ans. 10817.

(4.)

D

(5.) 10+2d2 + 2e3)÷(4a-4c-26)—(6+4d-2b) (6d-4c.) Ans. 411.

(6.) (8ab2-c2-d2) ÷ (3e2-d2)+12cd-8de.

288

Ans.

(1.) √b2 + √u2÷√d2 +2√d2÷4√a2 +6√a2 ÷

(2.)

5a35-3c1÷6+b÷c3÷d+d2÷b2. Ans.

92232 124

1024

(3.) (a+b)÷(d2+5d)+d÷2b++4d+2cd+d÷b

When a 50, b=32, c=16, d=8, e=2, f=0.

These are employed to include several terms which may sometimes be conveniently considered as one