Surface of Sphere.

The surface of a sphere equals four great circles.

TO FIND AREA WHEN DIAMETER IS GIVEN.

RULE.-Multiply the square of the diameter by 3.1416.

PROBLEMS.

Diameter of sphere equals 10 in.; what is area of sur

face? Ans. 314.16 sq. in.

2. Diameter of sphere equals 1 ft. 3 in.; what is area of surface? Ans. 4.91 sq. ft.

VOLUMES.

Prism or Cylinder.

TO FIND VOLUME WHEN AREA OF BASE AND ALTITUDE ARE GIVEN.

RULE.-Multiply the area of the base by the altitude.

PROBLEMS.

1. The edges of the base of a rectangular prism equal 2.2 in. and 1.1 in., the altitude 3.3 in.; what is the volume? Ans. 7.986 cu. in.

2. The edges of the base of a triangular prism equal 3 ft., 4 ft., and 5 ft., the altitude 10 ft.; what is the volume? Ans. 60 cu. ft.

3. The diameter of base of a cylinder equals 1 ft. 3 in., the altitude 2 ft. 6 in.; what is the volume? Ans. 3.067 cu. ft.

4. The circumference of the base of a cylinder equals 3.1416 ft., the altitude 3 ft. 9 in.; what is the volume? Ans. 2.945 cu. ft.

Right Pyramid or Cone.

TO FIND VOLUME WHEN AREA OF BASE AND ALTITUDE ARE

GIVEN.

RULE.-Multiply the area of the base by one-third of the

altitude.

PROBLEMS.

1. Each edge of the base of a right rectangular pyramid equals 2.5 ft., the altitude 2.25 ft.; what is the volume? Ans. 4.6875 cu. ft.

2. The radius of the base of a cone equals 5 ft., the altitude 21 ft.; what is the volume? Ans. 549.78 cu. ft.

Frustum, Right Pyramid, or Cone.

TO FIND VOLUME WHEN AREA OF BASES AND ALTITUDE ARE GIVEN.

RULE. To the sum of the two bases add the square root of their product, and multiply the result by one-third of the altitude.

PROBLEMS.

1. Each edge of the lower base of a frustum of a right quadrangular pyramid equals 3 ft., of the upper base 2 ft., the altitude 15 ft.; what is the volume? Ans. 95 cu. ft.

2. The diameters of the bases of a frustum of a cone equal 18 in. and 10 in., the altitude 16 in., what is the volume? Ans. 2,530.03 cu. in.

Sphere.

TO FIND VOLUME WHEN DIAMETER IS GIVEN.

RULE.-Multiply the cube of the diameter by .5236.

PROBLEMS.

1. The diameter of a sphere equals 10 in.; what is the volume? Ans. 523.6 cu. in.

2. The diameter of a sphere equals 15 in.; what is the volume? Ans. 1.0227 cu. ft.

CHAPTER XI

ALGEBRAIC EXPRESSIONS AND SIMPLE

EQUATIONS

ALGEBRAIC EXPRESSIONS.

An algebraic expression is a mathematical statement in which the quantities considered are represented by letters and the operations to be performed are indicated by signs.

The quantities may be either known or unknown. Known quantities are usually represented by the first letters of an alphabet; as a, b, c; a', b', c', read "a prime, b prime, c prime”; a", b", c", read "a second, b second, c second"; a1, b1, c1, read "a sub one, b sub one, c sub one," etc. Unknown quantities are usually represented by the last letters of an alphabet; as x, y, z; x', y', z'; X1, Y1, Z1; Xo, Yo, Zo, read "x sub zero, y sub zero, z sub zero," etc.

The signs are the same as those used in arithmetic.

Quantities multiplied together are called factors of their product. When factors are letters, the sign of multiplication

a C
b d

ac

bd'

is omitted; as aXbXc is written abc; X is written axaxaxaxb×b×b>c>c>d, is written ab3c2d and is read "a fourth, b cube, c square, d."

A coefficient is usually a number written before a quantity expressed by letters to show how many times the quantity is to be taken additively. Thus: in 24ab, 24 is the coefficient of ab, and shows that ab is taken, additively, 24 times.

When the quantity expressed by letters represents both known and unknown quantities, the product of the numerical coefficient and letters representing the known quantities is usually considered as the coefficient. Thus: in 7ax, 7a is regarded as the coefficient of x or 7 may be regarded as the coefficient of ax.

When no coefficient is expressed, the coefficient 1 is always understood. Thus: abc means the same as 1abc.

A term is an algebraic expression whose parts are not separated by a plus or a minus sign. Thus: 3ax, 5by, and 3ax÷5by are terms. In 2x2-3ax+4c2, 2x2-3ax, and +4c2, are, respectively, the 1st, 2d, and 3d terms.

Terms in algebraic expressions, like numbers in arithmetic, are divided into positive terms, or terms to be added together, and negative terms, or terms to be subtracted from positive terms, but added to negative terms. Positive terms are preceded by the sign +; negative terms by the sign term has no sign before it, it is considered positive.

When a

Like terms are terms which contain the same letters affected with the same exponents. Thus: 7a2x3 and 5a2x3 are like terms. When positive and negative like terms are combined, the result is called their algebraic sum.

Thus: 2a2x3 is the algebraic sum of 7a2x3 and −5a2x3.

Equal terms are like terms that have the same numerical coefficient. Thus: 7a2r3 and -7a2x3 are equal terms with unlike signs. When positive and negative equal terms occur in the same expression, they neutralize each other, or cancel.

The numerical value of an algebraic expression is the result obtained by assigning a numerical value to each letter and performing the operations indicated. Thus, the numerical value of

4a+3bc-d2,

when a = 1, b=2, c=3, and d=4, is 4X1+3×2×3-4×4=4+18-16=6.

PROBLEMS.

Find the numerical value of the following expressions, when a = 1, b=2, c=3, and d=4.

1. 3a2b2-2(a+d+1). Ans. 0.

when v = 1,435, p= 13.08, d=8, w=290;

also, when v=1,335, p=16.25, d=12, w= 800:

7. v-608.3(p+0.14d). Ans. approximately 0 and .8.

In dealing with the more simple algebraic expressions, the following rules apply:

ADDITION.

CASE I.-WHEN THE QUANTITIES TO BE ADDED ARE LIKE

TERMS.

RULE. Add the coefficients of the positive and negative terms separately; subtract the less sum from the greater, prefixing the sign of the greater; to the result annex the common literal part.