# Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/187

 14. The trisectrix $\rho = 2 a \cos \theta - a$. Ans. $R = \frac{ a(5 - 4 \cos \theta)^{\frac{3}{2}} }{ 9 - 6 \cos \theta }$ 15. The equilateral hyperbola $\rho^2 \cos 2 \theta = a^2$. Ans. $R = \frac{\rho^3}{a^2}$. 16. The conic $\rho = \frac{a(1 - e^2)}{1 - e \cos \theta}$. Ans. $R = \frac{a(1 - e^2)(1 - 2e \cos \theta + e^2)^{\frac{3}{2}}}{(1 - e \cos \theta)^3}$. 17. The curve $\begin{cases}x = 3 t^2, \\ y = 3 t - t^3.\ t = 1.\end{cases}$ Ans. $R = 6.$ 18. The hypocycloid $\begin{cases}x = a \cos^3 t, \\ y = a \sin^3 t.\ t = t_1.\end{cases}$ Ans. $R = 3 a \sin t_1 \cos t_1.$ 19. The curve $\begin{cases}x = a(\cos t + t \sin t), \\ y = a(\sin t - t \cos t).\ t = \frac{\pi}{2}.\end{cases}$ Ans. $R = \frac{\pi a}{2}.$ 20. The curve $\begin{cases}x = a(m \cos t + \cos mt), \\ y = a(m \sin t - \sin mt).\ t = t_0.\end{cases}$ Ans. $R = \frac{4 ma}{m - 1} \sin \left( \frac{m + 1}{2} \right) t_0.$

21. Find the radius of curvature for each of the following curves at the point indicated; draw the curve and the corresponding circle of curvature:

 (a) $x = t^2, 2 y = t; t = 1.$ (e) $x = t, y = 6 t^{-1}; t = 2$. (b) $x = t^2, y = t^3; t=1.$ (f) $x = 2 e^t,y = e^{-t}; t = 0.$ (c) $x = \sin t, y = \cos 2t; t= \frac{\pi}{6}.$ (g) $x = \sin t, y = 2 \cos t; t = \frac{\pi}{4}.$ (d) $x = 1 - t, y = t^3; t = 3.$ (h) $x = t^3, y = t^2 + 2t; t = 1.$

22. An automobile race track has the form of the ellipse $x^2 + 16y^2 = 16$, the unit being one mile. At what rate is a car on this track changing its direction

(a) when passing through one end of the major axis?

(b) when passing through one end of the minor axis?

(c) when two miles from the minor axis?

(d) when equidistant from the minor and major axes?

Ans. (a) 4 radians per mile; (b) $\frac{1}{16}$ radian per mile.

23. On leaving her dock a steamship moves on an arc of the semi cubical parabola $4 y^2 = x^3$. If the shore line coincides with the axis of y, and the unit of length is one mile, how fast is the ship changing its direction when one mile from the shore?

Ans. $\frac{24}{125}$ radians per mile.

24. A battleship 400 ft. long has changed its direction 30° while moving through a distance equal to its own length. What is the radius of the circle in which it is moving?

Ans. 764 ft.

25. At what rate is a bicycle rider on a circular track of half a mile diameter changing his direction?

Ans. 4 rad. per mile = 43' per rod.

26. The origin being directly above the starting point, an aëroplane follows approximately the spiral $\rho = \theta$, the unit of length being one mile. How rapidly is the aëroplane turning at the instant it has circled the starting point once?

27. A railway track has curves of approximately the form of arcs from the following curves. At what rate will an engine change its direction when passing through the points indicated (1 mi. = unit of length):

 (a) $y = x^3, (2, 8)$? (d) $y = e^x, x = 0$? (b) $y = x^2, (3, 9)$? (e) $y = \cos x, x = \frac{\pi}{4}$? (c) $x^2 - y^2 = 8, (3, 1)$? (f) $\rho \theta = 4, \theta = 1$?