EXERCISES - APPLIED PROBLEMS 1. The square of a certain number is 24 more than twice the number. Find the number. [HINT. Remember that there should be two solutions.] 2. The square of a certain number is less than of the number. Find the number. 3. The difference between the cube of a certain number and three times the square of the number is equal to four times the number itself. Find the number. [HINT. After forming the equation, divide both sides by x.] 4. The sum of the squares of two consecutive numbers is 1013. What are the numbers? [HINT. For the definition of consecutive numbers, see Ex. 36, p. 62.] 5. The product of two consecutive numbers is 272. What are the numbers? 6. The hypotenuse of a right triangle is 10 inches long and the sum of its sides is 14 inches. Find the lengths of the sides. 7. One side of a right triangle is 4 in. longer than the other. If the hypotenuse is 20 in. long, how long are the sides? [HINT. If x represent the shorter side, the equation here becomes x2+(x-4)2=400 and in solving this one of the solutions turns out to be negative. But a negative number can have no meaning in such an example as this, so we keep only the positive solution. This frequently happens in applied problems containing quadratics, so the pupil must always be on his guard to keep only such solutions as can actually fit a given example.] Ridge 8. An ordinary gable roof has the form (cross section) indicated in the figure. Suppose the "run" is 8 feet and the "" rafter 10 feet. How much greater would the "rise " be if an 11-foot rafter had been used? √57-6, or 1.54983+ ft. Ans. -Run Plate Rafter -Span of Roof- 9. A gardener spades a bed 40 feet long and 20 feet wide. He then decides to make the bed three times as large by adding to all sides a strip of the same width throughout. How wide must the strip be? 10. A rectangular plot of ground measures 160 feet by 40 feet. By how many feet must the length and breadth be equally increased so that the area becomes increased by 10,000 square feet?· 100√2-1, or 41.421+ ft. Ans. 11. A coach wishes to increase the length and breadth of a certain athletic field by the same amount in such a way that the diagonal line across the field will become increased by 50 feet. The field is now 400 feet by 300 feet. How many feet must be added to each dimension? 35.68+ ft. T Ans. 12. A circular swimming pool is surrounded by a walk 6 feet wide. The walk contains half as much area as the pool. Find (approximately) the radius of the pool. 13. If a train had its speed increased by 5 miles an hour, it could shorten its time for running 180 miles by 30 minutes. What is the rate in miles per hour? 14. The switchboard in a telephone office is an arrangement by which any one person who has a telephone may be connected with any other person in the system. If the number of persons in the system is n, it is known that the total number N of connections possible on the switchboard A A FIG. 82. is given by the formula N=n(n−1)/2. If 53,628 connections are possible, how many telephones are there in the system? 15. Figure 82 represents the gable end of a house with a circular window in it. The two points on the under side of the roof that are nearest the window are each 1 ft. from the outer circumference of the window frame; and they are 6 ft. from the under edge of the peak. If the diameter of the window frame is 6 ft., how far is its center beneath the under edge of the peak? 16. Figure 83 represents a pattern frequently used in window designs, consisting of a square ABCD with a semicircle EFG mounted upon it, the diameter GE of the semicircle being slightly less than one of the sides of the square. If the shoulders AG and DE are to be each 1 foot long and the total lighting surface is to be 88 square feet, find how long each side of the square must be made. 8 ft. Ans. B D C FIG. 83. 17. Solve Ex. 16 when the lower part of the window, instead of being a square, is to be a rectangle 3 feet higher than wide (other conditions remaining the same). 18. As the radius of a certain sphere was lengthened out 2 feet, the surface of the sphere became exactly double its original value. What was the original radius? 2(1+√2) ft. Ans. [HINT. The area of the sphere whose radius is r is 4 πr2.] 19. The figure represents a familiar form of pendant, or watch charm, consisting of a circular (or sometimes spherical) disk supported by two equal metal strips soldered to the circumference and meeting in a point above. Placing PT-t and calling the diameter of the circle d, show that the formula for the length of PS is PS=1(−d+√d2+4 t2). P T FIG. 84. Observe that by means of this formula the small length PS (which it is usually difficult to measure accurately) can be determined by measuring the larger and more accessible distances t and d. Explain how. FIG. 85. 20. The figure shows a rectangle whose base is 2 feet longer than its height. The rectangle is surrounded (circumscribed) by a circle. What must the dimensions of the rectangle be if the shaded area is to be 20 square feet? [HINT. The diagonal of the rectangle is a diameter of the circle.] PART III. SUPPLEMENTARY TOPICS † 158. The Graphical Solution of Quadratic Equations. Suppose we have the quadratic equation x2-3x-4=0. Let us represent the left member by the letter y; that is, let us write y=x2-3x-4. Now, if x is given some value, this equation determines a corresponding value for y. For example, if x=0, then y=02-3X0-4-4. Again, if x = 1, y = 12-3X1-4 -6. The table below shows a number of x values with their corresponding y values determined in this way: = This table is at once seen to resemble those in §§ 122, 124. Like them, it has a certain graph to correspond to it. This -6 -5 41 Y X 0 1 2 3 4 5 -1 2 3. 4 Y FIG. 86. X graph is obtained by first drawing axes XX and YY and then plotting (in the sense explained in § 122) each of the points x, y which the table contains and finally drawing the smooth curve which passes through all such points. The curve thus obtained is the graph of the given quadratic x2-3x-4=0. It is to be observed that the graph as thus determined is not a straight line and is therefore much different in character from the graph of a linear equation (compare § 123). And it is †This part contains a brief introduction to several topics which are considered in further detail in Vol. II. |