Question

Refer to Example 3.2–1. Use a surface plot and a contour plot of the perimeter length L as a function of d and θ over the ranges 1 ≤ d ≤ 30 ft and 0.1 ≤ θ ≤ 1.5 rad. Are there valleys other than the one corresponding to d = 7.5984 and θ = 1.0472? Are there any saddle points?

Answer 1

It is required to plot a surface plot and contour plot of the perimeter length L as a function of d and ?? over the ranges

1 ≤ d ≤ 30 feet

0.1 ≤ θ ≤ 1.5 rad

 

The function is:

L = 100/d – d/tan(θ) + 2d/sin(θ)

The MATLAB plot is given below. Firstly, the data points for d and theta is generated. Then, the mesh grid is created. Finally, length L is calculated at each data point.

 

Input:

% defining the data

d = 1:1:30;

th = 0.1:0.05:1.5;

[X, Y] = meshgrid(d, th);

L = 100./X - X./tan(Y) + (2.*X)./sin(Y);

% generating the surface and the contour plot

figure(1)

surf(X, Y, L);

xlabel(\'d\');

ylabel(\'sin(\\theta)\');

zlabel(\'Length\')

% generating the countour plot

figure(2)

contour(X, Y, L);

xlabel(\'d\');

ylabel(\'sin(\\theta)\');

 

Output:

SURFACE PLOT

 

CONTOUR PLOT

 

There are no other valley points and no saddle points.

There are no other valley points and no saddle points.