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296. When it is required to extract the cube root of a common fraction, or a mixed number, it is prepared in the same manner as directed in square root. (Art. 282.)
EXAMPLES FOR PRACTICE.
1. What is the cube root of 81% ? 2. What is the cube root of %? 3. What is the cube root of 4928? 4. What is the cube root of 1663 ? 5. What is the cube root of 85235?
· Ans. 196 Ans. 34. Ans. 51 Ans. 4.
APPLICATION OF THE CUBE ROOT.
297. The cube root may be applied in finding the dimensions and contents of cubes and other solids.
1. A carpenter wishes to make a cubical cistern that shall contain 2744 cubic feet of water; what must be the length of one of its sides?
Ans. 14 feet. 2. A farmer has a cubical box that will hold 400 bushels of grain ; what is the depth of the box? Ans, 7.92+ feet.
3. There is a cellar, the length of which is 18 feet, the width 15 feet, and the depth 10 feet; what would be the depth of another cellar of the same size, having the length, width, and depth equal ?
Ans. 13.92+ feet.
298. A Sphere is a solid bounded by one continued convex surface, every part of which is equally distant from a point within, called the center.
The Diameter of a sphere is a straight line passing through the center and terminated by the surface; as A B.
299. A Cone is a solid having a circle for its base, and its top terminated in a point, called the vertex.
296. How is a common fraction or a mixed number prepared for extracting the square root ? - 297. To what may the cube root be applied ?
298. What is a sphere? The diameter of a sphere ? — 299. What is a cone ?
The Altitude of a cone is its perpendicular hight, or a line drawn from the vertex perpendicular to the plane of the base; as B C.
300. SPHERES are to each other as the cubes of their diameters, or of their circumferences.
SIMILAR CONES are to each other as the cubes of their altitudes, or the diameters of their bases.
AU SIMILAR SOLIDS are to each other as the cubes of their homologous or corresponding sides, or of their diameters.
301. To find the contents of any solid which is similar to a given solid.
RULE. — State the question as in Proportion, and cube the given sides, diameters, altitudes, or circumferences, and the fourth term of the proportion is the required answer.
302. To find the side, diameter, circumference, or altitude, of any solid, which is similar to a given solid.
RULE. State the question as in Proportion, and cube the given sides, diameters, circumferences, or altitudes, and the cube root of the fourth term of the proportion is the required answer.
EXAMPLES FOR PRACTICE. 1. If a cone 2 feet in hight contains 456 cubic feet, what are the contents of a similar cone, the altitude of which is 3 feet?
Ans. 1539 cubic feet. OPERATION.
23 : 33 :: 45 6:15 3 9 2. If a cubic piece of metal, the side of which is 2 feet, is worth $6.25, what is another cubical piece of the same kind worth, one side of which is 12 feet?
Ans. $ 1350. 3. If a ball, 4 inches in diameter, weighs 50lb., what is the weight of a ball 6 inches in diameter ? Ans. 168.7+1b.
299. What is the altitude of a cone? 300. What proportion do spheres have to each other? What proportion do cones have to each other? What proportion do all similar solids have to each other ? — 301. What is the rule for finding the contents of a solid similar to a given solid ? - 302. The rule for finding the side, diameter, &c., of a solid similar to a giveu solid?
4. If a sugar-loaf, which is 12 inches in hight, weighs 161b.,
be broken from the base, that the residue may weigh 8lb.?
Ans. 2.5+ in. 5. If an ox, that weighs 800lb., girts 6 feet, what is the weight of an ox that girts 7 feet ?
Ans. 1270.3lb. 6. If a tree, that is 1 foot in diameter, make 1 cord, how many cords are there in a similar tree, whose diameter is 2 feet?
Ans. 8 cords. 7. If a bell, 30 inches high, weighs 1000lb., what is the weight of a bell 40 inches high?
Ans. 2370.3lb. 8. If an apple, 6 inches in circumference, weighs 16 ounces, what is the weight of an apple 12 inches in circumference ?
Ans. 128 ounces. 9. A and B own a stack of hay in a conical form. It is 15 feet high, and A owns of the stack; it is ed to know how many feet he must take from the top of it for his share.
Ans. 13.1+ feet.
303. Arithmetical Progression is a series of numbers that increases or decreases by a constant difference.
The Terms of the series are numbers of which it is formed.
The Common Difference is the constant difference between the terms.
The series is ascending when each term after the first exceeds that before it, and descending when each term after the first is less than that before it.
Thus, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, is an ascending series, and 29, 26, 23, 20, 17, 14, 11, 8, 5, 2, is a descending series.
303. What is arithmetical progression ? What are the terms of a progression ? The extremes ? The means ? Common difference? What is an ascending series? A descending series ?
In Arithmetical Progression, the first term, the last term, the number of terms, the common difference, and the sum of the terms, are so related to each other, that any three of these being given, the two others may be readily determined.
304. To find the COMMON DIFFERENCE, the first term, last term, and number of terms being given.
ILLUSTRATION. - In the following series,
2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 2 and 29 are the extremes, 3 the common difference, 10 the number of terms, and the sum of the series 155.
It is evident that the number of common differences in any series must be 1 less than the number of terms. Therefore, since the number of terms in this series is 10, the number of common differences will be 10—1=9, and their sum will be equal to the difference of the extremes; hence, if the difference of the extremes (29 – 2= 27) be divided by the number of common differences, 9, the quotient, 3, will be the common difference. Hence the
RULE. — Divide the difference of the extremes by the number of terms less one, and the quotient will be the common difference.
EXAMPLES FOR PRACTICE.
1. The extremes of a series are 3 and 35, and the number of terms is 9; what is the common difference ?
2. If the first term is 7, the last term 55, and the number of terms 17, required the common difference.
Ans. 3. 3. If the first term is 4, the last term 14, and the number of terms 15, what is the common difference?
4. If a man travels 10 days, and the first day goes 9 miles, and the last 17 miles, and increases each day's travel by an equal difference, what is the daily increase ?
Ans. f miles.
303. What five things are named, any three of which being given the other two can be found ? - 304. The rule for finding the common difference, the first term, last term, and number of terms being given ?
305. To find the SUM OF ALL THE TERMS, the first term, last term, and number of terms being given.
ILLUSTRATION. Let the two following series be arranged as follows:
2, 5, 8, 11, 14, 17, 20, 77, sum of first series. 20, 17, 14, 11, 8, 5,
77, sum of inverted series. 22, 22, 22, 22, 22, 22, 22, 154, sum of both series.
From the arrangement of the above series, we see that, by adding the two as they stand, we have the same number for the sum of the successive terms, and that the sum of both series is double the sum of either series.
It is evident that, if 22 in the above series be multiplied by 7, the number of terms, the product will be the sum of both series ; thus, 22 X 7 · 154; and, therefore, the sum of either series will be 154 ; 2 77. But 22 is the sum of the extremes in each series ; thus, 20 + 2 22. Therefore, if the sum of the extremes be multiplied by the number of terms, the product will be double the sum of either series. Hence,
RULE 1. Multiply the sum of the extremes by the number of terms and half the product will be the sum of the series. Or,
RULE 2. Multiply the sum of the extremes by half the number of terms, and the product will be the required sum.
EXAMPLES FOR PRACTICE.
1. If the extremes of a series are 5 and 45, and the number of terms 9, what is the sum of the series ?
( 45 + 5) X9
225, sum of the series.
2 2. John Oaks engaged to labor for me 12 months. For the first month I was to pay him $ 7, and for the last month $ 51. In each successive month he was to have an equal addition to his wages ; what sum did he receive for his year's labor ?
Ans. $ 348.
305. The rule for finding the sum of all the terms, the first term, last term, and number of terms being given ?