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EXERCISES — REVIEW OF CHAPTER I
I. ORAL EXERCISES 1. If k represents a certain number, what represents 12 times that number?
2. If a yard of cloth costs m dollars, what will x yards cost?
4. If a train goes y miles per hour, how fast does it travel per minute?
5. In how many hours can a man walk x miles if he goes at the rate of y miles per hour?
6. A man has a dollars and b quarters; how many cents has he ? 7. How many dimes are there in x dollars and y half-dollars ?
8. I have x dollars and y dimes. If I spend 60 cents, how much (in cents) have I left?
9. If y represents a certain number, what represents 6 less than three times y?
10. By how much does 25 exceed x ?
12. What is the cost of 2 dozen oranges at n cents per dozen and 3 dozen lemons at m cents per dozen ?
13. If a yard be divided into x equal parts, how many inches will there be in each part?
14. If I stands for the length in feet of a running track, what is the length (in feet) of a track 50 feet longer? What is the length of a track 100 yards shorter ?
15. If I am 3 years old now, how old was I y years ago? How old will I be c years hence ?
16. A man sold an automobile for $500 and lost a dollars. What did the automobile cost ?
17. It costs 3 cents for each ounce or fraction thereof to send a letter through the mail. How much does it cost to send a letter that weighs a fraction more than n ounces?
II. WRITTEN EXERCISES 18. A rule stated in letters is called a formula. For example, the area of a triangle equals one half the product of the base by the altitude. Stated as a formula, this becomes
A=ibh, where A stands for area, b for base, and h for altitude (or height).
By use of this formula find the area of the triangle whose base is 6 inches and
Fig. 7. whose altitude is 3 inches.
[Hint. Substitute (or place) the given values of b and h into the formula and see what value results for A.]
19. The gable of a certain house is a triangle whose base measures 35 feet and whose altitude is 18 feet. Find, by the formula of Ex. 18, how many square feet of lumber it contains.
20. It is shown in Geometry that Fig. 8.
“the square drawn on the hypotenuse of a right triangle is equal to the sum of the squares drawn on the other two sides.” Express this rule in a formula, using h for hypotenuse, x for one side, and y for the other side. 21. Find the area of the square
Triangle on the hypotenuse of a right triangle if the sides are 6 feet and 10 feet in length; if the sides are 15 inches and 22 inches in length.
[Hint. Use the formula you obtained in Ex. 20.)
22. The circumference of a circle is expressed by the formula
C=2 ur, where C stands for circumference, r for radius, and 7 (read pi) stands for the number 3.1416 (usually taken as 37).
By means of this formula find the cirFig. 10.
cumference of the circle whose radius is 8
inches. Do the same when the radius is i of an inch, when it is 2} feet, and finally when the diameter is 4 feet.
23. If my bicycle wheel has a diameter of 24 inches, how far does the bicycle go in one turn of the wheel ?
[Hint. The distance moved=the length of the circumference.] 24. The formula for the radius of a circle is
where r stands for the radius, C for the circumference, and a has the value mentioned in Ex. 22. Write out (in words) the rule thus expressed.
25. The formula for the area of a circle is A=ar2, where r stands for the radius, A for the area, and a for the same number as in Ex. 22.
Write (in words) the rule thus expressed and find the area of the circle whose radius is 3 inches. Find the area also of the circle whose diameter is 28 inches.
26. A farmer builds a cylindrical silo which has for its base a circle 24 feet in diameter. How many square feet in the base? In working this, first use the value 37 for a, then use the more accurate value 3.1416. By how much do your two results differ?
Fig. 11.- Silo.
27. The volume of a cylinder is the product of its height and the area of its base. Find the volume of the entire silo of Ex. 26, if its height is 50 ft.
Find the amount of material in the silo when it is full to a height of 15 ft. ; 20 ft.; x ft. 28. The volume of a sphere is expressed by the formula
V=fari, where r stands for the radius (AO in Fig. 12), V for the volume, and a for the same number as in Ex. 22. By means of this formula, find the volume of a sphere whose radius is 3 inches.
Find the volume of a sphere whose radius is 4 feet.
Find the volume of a sphere whose diameter is 9 feet.
29. By the "gear" of a bicycle is meant the diameter of one of the wheels multiplied by the number of teeth in
P the front sprocket wheel divided by
Fig. 12. the number of teeth in the rear sprocket wheel. Express this rule in a formula, using the letter g for gear, D for diameter of wheel, T for number of teeth in front
sprocket wheel, and t for number of teeth in rear sprocket wheel.
30. Use the formula you found in Ex. 29 to find the gear of the bicycle whose wheel meas
ures 28 inches in diameter and o whose front and rear sprocket
wheels have 18 and 7 teeth, reFig. 13.
spectively. Try some other combinations to show the effect on the gear of changes in the number of teeth on the sprocket wheels.
31. In Fig. 14, one wheel is turning another by means of a belt.
D=the diameter of the large wheel,
n=the number of turns it is making per minute, then during the motion we always have
DN=dn. By means of this formula answer the following question: If the larger wheel is 1 foot in diameter and is making 10 turns (revolutions) per minute, how many turns per minute is the smaller wheel making if its diameter is 3 inches ?
32. In the figure an engine is turning the central part (armature) of a dynamo. The armature wheel (which the belt runs
over) is 1 foot in diameter, while the engine wheel (driving wheel) is 6 feet in diameter.
How many revolutions must the driving wheel have per minute in order that the armature may revolve 1200 times per minute?