EXERCISES

1. Find the area and volume of a sphere whose radius is 5 inches.

2. Find the volume of the sphere inscribed in a cone of revolution whose altitude is 12 inches, the radius of whose base is 5 inches.

3. For what values of the radius is the number of units in the volume of a sphere less than the number of units in the area? For what values of the radius is DiV <DtS? DiV>DS? In biology, we learn that a cell subdivides after a certain time. Why?

4. The diameter of a sphere is found by measurement to be 8.3 inches with a probable error of 0.1 inch. What is the error in the computed value of the area of the sphere? In the computed value of the volume? Is the percentage error in the area greater or less than the percentage error in the volume?

5. What is the relation between the rates of increase of the radius and volume of a soap bubble? When the radius is 3 inches and is increasing at the rate of 0.1 inch per second, how fast is the volume increasing? The surface?

6. A washbowl is in the form of a hemisphere of radius 8 inches. How much water is there in it when the water is 5 inches deep?

7. When the depth of the water in the bowl of Exercise 6 is 4 inches the surface is falling at the rate of of an inch per second. How rapidly does the water run off through the drain pipe?

8. Find the area and volume of the earth, assuming that it is a sphere whose radius is 3957 miles. Find the error and relative error in each case if the error in the radius is not more than 7 miles. How does the relative error in the area and in the volume compare with the relative error in the radius?

9. How many lead shot of an inch in diameter can be made from a piece of lead pipe 2 feet long whose outside and inside diameters are respectively 1.25 inches and 1 inch?

curves y

MISCELLANEOUS EXERCISES

1. Find the volume generated by revolving the area bounded by the x2 and x = y2 about the x-axis. 2. A ball rolls down a smooth plane inclined at an angle of 18°. Find how far it will roll in t seconds.

3. The diameter of a cone is found to be 5 inches, and the altitude 8 inches. If the error in the diameter is 0.06 of an inch and the altitude is exact, find the error in the computed value of the lateral area. DEFINITION. At the center O of an equilateral triangle ABC erect a line perpendicular to the plane of the triangle. Take a point D on it such that AD = AB, and join D to A, B, and C. The figure so obtained is called a regular tetrahedron [Fig. 193 (a)].

4. Find the altitude DO and the volume of a regular tetrahedron (a) whose edge AB is 6 inches; (b) whose edge is e.

DEFINITION. At the center O of a square ABCD erect a line perpendicular to the plane of the square, and extend it on both sides of the plane to points E and F such that EA FA AB. Join E and F to the vertices of the square. The resulting figure is called a regular octahedron.

5. Find the volume of a regular octahedron (a) whose edge AB is 6 inches; (b) whose edge is e. (Note that EF is a diagonal of the square AECF.)

6. Express the total area of the octahedron in terms of the edge. Do the same for the cube and the regular tetrahedron, and plot the graphs of these three functions of e on the same axes. For the same value of e, which solid has the greatest area? Which area increases the most rapidly as e increases?

7. The graph of x + y1 – a‡ is a parabola tangent to the coördinate axes whose axis of symmetry bisects the first quadrant. Find the area bounded by the curve and the axes.

8. Find the volume of the solid generated by revolving about the y-axis the area bounded by the parabola y = ax2, x = 0, and y = h. Show that this volume is one-half the volume of a cylinder whose altitude is h and whose bases are equal to the circle forming the upper surface of the solid. 9. A point moves so that its coördinates at any time t are given by

Find the components parallel to the axes of the velocity and acceleration. Show that the point moves in a circle (to eliminate t, square both equations

and add the results) and determine the part of the circle it describes from t = 0 to t 1. Find the velocity and acceleration (direction and magnitude) at these times

=

10. A water resevoir is in the form of the figure generated by revolving the curve 12y = x2 about the y-axis. How much does it contain if the water is 10 feet deep at the center? If water is entering at the rate of 10 cubic feet per second when the depth at the center is 12 feet, how rapidly is the surface rising?

CHAPTER VIII

PROPERTIES OF TRIGONOMETRIC FUNCTIONS

LOGARITHMIC SOLUTION OF TRIANGLES, CASES III AND IV 116. Introduction. In this Chapter we shall derive formulas which express properties of the trigonometric functions analogous to certain properties of algebraic, exponential, and logarithmic functions with which we are already familiar. These properties enable us to change the form of expressions involving trigonometric functions, a process of great importance in those parts of more advanced mathematics where these functions appear. These properties, together with the formulas for the functions of n90° ± 0 (page 192), are used more, perhaps, than the solution of triangles, except in such fields as surveying.

As an application of these formulas we shall obtain formulas for the solution of triangles, which are adapted to the use of logarithmic tables, for those cases in which we have hitherto used the law of cosines.

In the sections immediately following we shall consider the interdependence of the trigonometric functions of an angle 0, with applications to the solution of equations and the proof of identities.

117. Fundamental Trigonometric Relations. As an immediate consequence of the definitions of the trigonometric functions we obtained the reciprocal relations

in which sin20 denotes the square of sin 0, or (sin 0)2, and cos2 denotes (cos 0)2.

To prove (5), we have, for any value of 0,

(4)

(5)

EXAMPLE. Express each of the trigonometric functions in terms of sin 0.

The sign of the radical must be chosen from a knowledge of the quadrant in which lies.

These results may also be obtained from a figure by means of the definitions of the functions (see Exercises 9 and 10, page 170). The given x function, sin 0, may be regarded as a fraction with unit denominator. Describe a circle 1, and draw the line y = sin 0, regarding Then the angles

about the origin with radius r

Ø as constant. Let the line cut the circle in P and P'.