« ΠροηγούμενηΣυνέχεια »
ART. 441.-Comparison of Similar Solids.
Principles.-1. The contents of similar solids are to each other as the cubes of their corresponding dimensions.
2. The corresponding dimensions of similar solids are to each other as the cube roots of their contents.
1. The side of a cubical block is 4 inches and the side of another is 6 inches : how many times is the bulk of the second greater than that of the first?
2. If a globe of gold 2 inches in diameter is worth $900, what is the value of one 5 inches in diameter ?
3. The diameters of two spheres are respectively 3 and my inches : how many times the volume of the smaller sphere is the larger ?
4. If a man 6 feet high weighs 144 pounds, what is the weight of a boy of similar build whose height is 4 feet ?
5. A stack of hay 14 feet high contains 9 tons : how high must a similar stack be to contain 36 tons ?
6. What is the diameter of a ball whose contents equal that of two balls that are respectively 4 and 5 inches in diameter ?
7. If a cylindrical cistern 5 feet in diameter contains 24 hogsheads of water, how much will a similar cistern contain whose diameter is 18 feet ?
8. A cubical box contains 343 cubic inches : what are the dimensions of the box inside ?
9. The side of a cubical vessel is 2 feet : find the side of another cubical vessel that shall contain 5 times as much.
10. How many balls 5 inches in diameter equal in volume a ball whose diameter is 20 inches ?
11. Illustrate Art. 441 by an original problem.
ART. 442.-An Arithmetical Progression is a series of numbers that increase or decrease by a constant difference; as 2, 4, 6, 8, and 12, 10, 8, 6, 4, 2.
ART. 443.—The Terms of a series are the numbers that compose it.
ART. 444.—An Ascending Series is one in which each term is greater than the preceding term ; as 3, 5, 7, 9, 11, etc.
ART. 445.—A Descending Series is one in which each term is less than the preceding term; as 15, 12, 9, 6, 3.
ART. 446.—The Extremes are the first term and the last term.
ART. 447.—The Means are the terms between the extremes.
ART. 448.—The Common Difference is the difference between two consecutive terms.
ART. 449.—The Sum of the Terms is the number obtained by adding together all the terms.
ART. 450.—In an arithmetical series five elements are concerned : the first term, the last term, the common difference, the number of terms, and the sum of the series. When any three of these are given, the others may be found.
1 If the first term of an ascending arithmetical progression is 2, the common difference 2, and the number of terms 12, what is the last term ?
Solution. The statement of this question shows that 2, the common difference, is to be added 11 times to the first term in order to make the 12th or last term. Hence, the last term is 2 + (2 x 11), which is 24.
2. If the first term of a descending series is 30, the common difference 3, and the number of terms 10, what is the last term ?
Solution.—In this question it is clear that the common difference, 3, is subtracted 9 times from the first term in order to make the last term The last term then is 30 – (9 x 3), that is 30 – 27 = 3:
3. If the last term is 60, the common difference 3, and the number of terms 20, what is the first term ?
Solution. In order to make 60 the last term, the common difference, 3, must have been added to the first term 19 times. 3, taken 19 times is equal to 57, which, in order to make 60, must be added to the difference between 60 and 57, or 3 : hence, 3 is the first term.
4 If the first term is 5, the last term 80, and the number of terms 26, what is the common difference ?
Solution.—The statement of this question shows that the first term has been increased to the extent of 75, by 25 equal additions : consequently each addition was 75 – 25 3, which must be the common difference.
5 If the first term is 4, the last 94, and the common difference 5, what is the number of terms ?
Solution.—The first term, 4, has been increased to the extent of 90 by adding the common difference, 5, a certain number of times. How often must 5 be added in order that it may amount to 90 ? Evidently 90 = 5, or 18, is the number of terms exclusive of the first : the number of terms, therefore, is 19.
6. If the first term is 4, the last term 25, and the number of terms 8, what is their sum ? Solution. The common difference found as in example 4, is 3,
4 ng 10 13 16 19 22 25 25 22 19 16 13 10 7 4
29 29 29 29 29 29 29 29 Writing out the series in direct and in reverse order and adding the two, we obtain 29 as many times as there are terms. That is, twice the sum of the series is 29 x 8 = 232. Hence, 232 +2, or 116, is the sum of the series,
It follows, therefore, that the sum of a series equals the sum of the extremes multiplied by half the number of terms.
7. If the first term of a decreasing series is 100 and the common difference 3, what is the 32d term ?
8. If John sold 25 apples at the rate of half a cent for the first, 1 cent for the second, 13 cents for the third and so on, what price was received for the last apple ?
9. Caspar walked 10 days, traveling 3 miles farther each day than on the preceding day. On the last day he walked 33 miles : how far did he walk the first day?
Geometrical Progression. ART. 451.–A Geometrical Progression is a series of numbers that increase or decrease by a common multiplier; as, 3, 9, 27, 81 or 36, 12, 4.
ART. 452.—The Ratio is the common multiplier ; thus, in the foregoing series the rate is 3 and } respectively. In a descending series the ratio is a fraction.
WRITTEN EXERCISES. 1. If the first term is 3, the rate 4 and the number of terms 6, what is the last term ?
Solution.—The second term 3 x 4; the third = 3 x 42 ; the fourth = 3 x 43, etc. That is, the last term equals the first term multiplied by the ratio raised to a power whose exponent is one less than the number of terms. Hence, the 6th term of the series equals 3 x 45 = 3 x 1024 = 3072.
2. If 3 is the first term, 4 the rate, and the number of terms 5, what is the sum of the terms ? Solution. The series written out is
3, 12, 48, 192, 768. Multiplying by 4, the ratio, and reversing terms, 2072, 758, 192, 48, 12. Subtracting original series reversed,
768, 192, 48, 12, 3. Gives (series x 4) series series x 3
3069. Since 3 times the series = 3069, 3069 = 3, or 1023, = sum of series.
Hence, the sum of a series is found by multiplying the last term by the ratio, and dividing the difference between the product and the first term by the ratio less 1.
3. The first term is 4, the rate 3: what is the sixth term ?
4. The first term of a descending series is 48, and the second 24: what is the 8th term ?
5. The first term is 5, the rate 5 and the last term 15,625 ; what is the sum of the terms ?
6. A man accepted an offer to buy 15 dictionaries, on the condition that he should pay 2 cents for the first volume, 4 cents for the second, 8 cents for the third, and so on; what did he owe for the 15 books ?
7. A father agreed to deposit 1 cent in a savings bank for his son when he was a year old, and to double the deposit on each succeeding birthday until the son was 21 years of age. Find the sum of the deposits.
NOTE,-Few realize the enormous increase of a number when expanded by geometrical ratio. If a man should agree to buy a horse on the condition that he paid 1 cent for the first nail in its shoe, 2 cents for the second, 4 cents for the third, and at that rate until the whole 32 nails had been paid for, the sum required (as the pupil can learn) would be over $40,000,000.
square root ?
REVIEW QUESTIONS. What is meant by the square or second power of a number ? The cube? What is an exponent ? The root or base of a number ? What is involution ? Evolution ? The root of a number ? The
What is the index of a root ? What is evolution ? Give the rule for the extraction of the square root of a number. How do the areas of similar figures compare with each other? How do the corresponding dimensions of similar figures compare ?
Give the rule for the extraction of the cube root of a number. How do similar volumes compare with each other? How do the corresponding dimensions of similar volumes compare ?
What is an arithmetical progression ? An ascending series ? A descending series ? What are the terms of a series? The extremes ? The means ? What is the common difference? The sum of the terms ? What is a geometrical progression ? The ratio ?