Understanding Engineering Mathematics

7.4 Applications

1.In simplelinear programmingwe consider a set of linear relations that constrain two
variablesx,yin the form
ax+by≤c

and look for the maximum or minimum value of some linear function

f(x, y)=mx+ny

asxandyvary subject to these constraints. By considering the equations of straight

lines, show how to solve such problems graphically and by solving simultaneous equa-

tions. Find the values ofxandysuch that

x≥ 0 ,y≥ 0

4 x−y− 24 ≥ 0

x−y+ 7 ≥ 0

f(x, y)=x+yis a maximum

2.This question considers a very simple model of the ‘slingshot effect’, in which a space
module is transferred from an orbit around the earth to one around the moon, say. The
trick is to eject the module from the earth orbit with just the right speed and direction
for it to travel in a straight line to intercept the moon orbit at just the right point and
speed to be captured and enter into orbit around the moon. Figure 7.11 shows how
this can be modelled by finding the straight line that is tangent to the earth and moon
orbits represented as circles centre the origin, radiusRand centre (0,D), radiusr,
respectively. HereDis the earth moon distance,Rthe earth orbit radius andrthe
moon orbit radius. Find the equation of this straight line, the distance from orbit to
orbit, and the points where the orbits are left and entered all in terms ofD,R,r.Try
out your results on some realistic data.

0 x

y

r

R

D

Figure 7.11