Calculus: Analytic Geometry and Calculus, with Vectors

302 Functions, graphs, and numbers

feet if r > s and this number is positive. Investigation of the whole matter is
not as simple as might be supposed. Hint: If x > 0 and the man runs from the
origin to the point (x,0), we should be able to calculate (in terms of x and the
given constants) the distance he runs, the distance he swims, and the total time
T required to reach the island.
27 Light travels with speed s, in air and with speed S2 in water. Figure
5.297 can interest us in possible paths by which
___C __, light might journey from a point 11 in the air to

Figure 5.297

a point S on the surface of the water and then

to a point W in the water. Show that the total

time T is a minimum when the point S is so

situated that the angle 01 of incidence and the

angle 02 of refraction satisfy the condition

sin 01S1 sin 02S2

Remark: The above formula is the Snell formula, one of the fundamental formulas

of optics. Phenomena such as the one revealed by this problem are of great

interest in physics and philosophy.

28 As in Figure 5.298 a heavy object of weight W is to be held by two identical

x

Cable Cable

Figure 5.298

cables. A kind engineer tells us that the tension

B T in the cables is W a2 + x2/2x. A solemn

merchant tells us that the cost per foot of his

cables is kT dollars, where T is the tension they

will safely withstand. We must buy the cables,

and we have a problem. Ans.: We buy 2 a

feet of cable costing Wk// dollars per foot, so

we need 2Wka dollars.

29 Modify the preceding problem by supposing that the body must hang

below the point which lies between !1 and B at unequal distances a and b from

and B.

30 The lower free corner of a page of a book is folded up and over until it

meets the inner edge of the page and then the folded part is pressed flat to leave

a triangular flap and a crease of length L. Supposing that the page has width a,

find the distance from the inner edge of the page to the bottom of the crease when

L is a minimum and find the minimum L. Hint: Minimize V. An:.: a/4 and

3 ' a/4.

31 Sketch the part G1 of the graph of the equation

y=x+1

that lies in the first quadrant and observe that the y axis and the line having

the equation y = x are asymptotes of G1. Someday we will learn that G1 is a

branch of a hyperbola and that the point V on G1 closest to the origin is a vertex

of the hyperbola. Find the coordinates of F and the distance from the origin

to V. Ans.: (2-i', (1 + /)2-34) and V2 2

32 A given circle has radius a. A second circle has its center on the given

one, and the arc of the second circle which lies inside the given circle has length L.