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slipping on the circumference of a fixed circle is called an epicycloid or a hypocycloid according as the rolling circle touches the outside or inside of the fixed circle. If the tracing point is not on the circumference of the rolling circle but on a radius or radius produced, the curve it describes is called a trochoid if the circle rolls upon a straight line, or an epitrochoid or a hypotrochoid if the circle rolls upon another circle. These curves will be discussed in the calculus.

Exercises

1. Construct a cycloid by dividing a generating circle of radius 1.15 inches into twenty-four equal arcs and dividing the base into, intervals 3/10 inch each.

2. Compare the cycloid of length 2π and height 1 with a semiellipse of length 27 and height 1.

3. Write the parametric equations of a cycloid for origin C, Fig. 160.

4. Write the parametric equations of a cycloid for origin B, Fig. 160. 5. Find the coördinates of the points of intersections of the cycloid with the horizontal line through the center of the generating circle. 6. Show that the top of a rolling wheel travels through space twice as fast as the hub of the wheel.

7. By experiment or otherwise show that the tangent to the cycloid at any point always passes through the highest point of the generating circle in the instantaneous position of the circle pertaining to that point.

Exercises for Review

1. Simplify the product:

(x − 2 − √3) (x − 2 − i √3) (x − 2 + √3) (x − 2 + √ï3).

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3. Find tan by means of the formula for tan (A + B), if 0 = tan-1 1/2+tan-1 1/3.

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5. Find the equation of a circle whose center is the origin and which passes through the point 14, 17.

6. The first of the following tests was made in 1875 with the automatic air brake on a train composed of cars weighing 30,000

pounds. The second in 1907 with the "LN" brake on a train composed of cars weighing 84,000 pounds. Find by use of logarithmic paper the equation connecting the speed and the distance run after application of the brakes.

Distance run after application of brake

Corresponding speed

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9. A fixed point located on one leg of a carpenter's "square" traces a curve as the square is moved, the two arms of the square, however, always passing through two fixed points A and B. Find the equation of the curve.

10. Find the parametric equations of the oval traced by a point attached to the connecting rod of a steam engine.

11. The length of the shadow cast by a tower varies inversely as the tangent of the angle of elevation of the sun. Graph the length of the shadow for various elevations of the sun.

12. From your knowledge of the equations of the straight line and circle, graph:

y = ax + √a2 − x2.

(See Shearing Motion, §37.)

13. In the same manner, sketch:

y = a + x + √a2 — x2.

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Has this curve a minimum value for all positive values of a and b?

15. Find by use of logarithmic paper the equations of the curves of Fig. 163. These curves give the amounts in cents per kilowatthour that must be added to price of electric power to meet fixed charges of certain given annual amounts for various load factors.

16. The angle of elevation of a mountain top seen from a certain point is 29° 4'. The angle of depression of the image of the mountain

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FIG. 163.

Annual Fixed Charges of $10, $15 and $20 of an Hydroelectric Plant, Reduced to Cents per kw-hour for Various Load Factors.

top seen in a lake 230 feet below the observer is 31° 20'. Find the height and horizontal distance of the mountain top, and produce a single formula for the solution of the problem.

17. Find the points of intersection of the curves:

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21. State the remainder theorem and illustrate by an example.

22. Find the compound interest on $1000 for twenty-five years at 5 per cent. Show how to solve by means of progressions.

23. The curve y2 quadrants is y1

=

=

x1 appears in which quadrants? In what x6? Compare the curves x3y2

=

1 and x2y3

=

1. 24. Which trigonometric functions of increase as increases in the first quadrant? Which decrease?

25. Given sin 30°

=

1/2, cos 45° √1/2. Find the following:

=

sin 150°, cos 135°, sin 225°, cos 300°, sin 330°, sin ( 30°).

26. Which is greater, tan 7° or sin 7°, and why? Which is greater,

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28. From the graph of y = x2 obtain the graph of 4y

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= x2 and of

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Meters.

FIG. 164. Trajectory of a German Army Bullet for a Range of 1000

29. Given, cos 0

=

2/5; find sin 0, tan 0 and cot 0.

30. Find the equations of the six straight lines determined by the intersections of:

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31. In Fig. 164 the full drawn curve is the trajectory of the projectile of a German Army bullet for a range of 1000 meters. The dotted curve is the theoretical trajectory that would have been described by the bullet if there had been no air resistance. The dotted

curve is a parabola (of second degree). Find its equation, taking the necessary numerical data from the diagram.

32. Find the maximum value of Ρ if p = 3 cos 0 4 sin 0.

33. Find the maximum value of y if y = find the value of x for which y is a maximum.

√3 cos x

- sin x, and

34. In Fig. 165 let ABCO be a square of side a. Show that for all positions of ON, CM × AN

=

a2, and hence show how to use

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FIG. 165.-Construction of a Constant Product CM × AN = AB2.

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