Computer Algebra Recipes

194 CHAPTER 4. NONLINEAR ODE MODELS
TheabsolutevalueofY was plotted, because in some executions of the work
sheet Y will be negative because the ordering of the answers in sol is reversed.
The shape of the strip that maximizes the area is a circular arc.
The area R a
Ydxis now calculated, the absolute value being taken to avoid
0
a possible negative area if the \wrong" solution is selected in sol.
> A:=abs(int(Y,x=0..a));
A := 0:3572686358
The maximum area is about 0:36 square milions. Without doing any mathematical
calculation, can you suggest why this is a maximum and not a minimum?
PROBLEMS:
Problem 4-39: Maximum volume of a solid
Acurvey(x) of length 2 is drawn between the points (0, 0) and (1, 0) in such
a way that the solid obtained by rotating the curve about the x-axis has the
largest possible volume.
(a) Determine y(x).
(b) Plot y(x) over the range x =0to1.
(c) What is the value of y at x =0:5?
(d) Make a three-dimensional plot of the solid.
(e) What is the volume of the solid?
Problem 4-40: The catenary curve
Consider a uniform cable of length L =1:5 km and mass per unit length ² =1
kg/m suspended between the two endpoints (¡a=2, b) and(a=2, b), where
a = b =1km.
(a) Determine the equilibrium shape (referred to as a catenary curve) of the
cable. Hint: The potential energy of the cable will be at a minimum when
the cable has its equilibrium shape. Take g =10m/s2 .
(b) Plot the equilibrium shape of the cable.
(c) If the cable crosses a very deep Himalayan gorge with the river located a
distance b below the endpoints of the cable, what is the distance between
the minimum in the cable and the river?
(d) What is the distance down to the river from a point one-third of the way
along the cable?
(e) What force is exerted on the supports at the endpoints of the cable?
(f) What length of cable should be used if the sag in the middle of the cable
is not to exceed 50 m?