Verified answer

Pick the stunningly original notation that she invests a and b in type A and B bonds respectively.

>"wants to invest up to $20,000"

a+b ≤ 20,000

>"She can purchase a type A bond yielding a 10% return on the amounted invested, and she can purchase a type B bond yielding a 15% return on the amount invested."

Return(a,b) = 0.1a + 0.15b

>"She wants to invest at least as much in the type A bond as in the type B bond."

a ≥ b

>"She will also invest at least $5000 in the type A and no more than $8000 in the type B bond."

a ≥ 5000

b ≤ 8000

>"How much should she invest in each type of bond to maximize her return?"

So you want to use linear optimization find the maximum of

Return(a,b) = 0.1a + 0.15b

on the region constrained by these constraints:

a ≥ b [Constraint 1]

a+b ≤ 20,000 [Constraint 2]

a ≥ 5000 [Constraint 3]

b ≤ 8000 [Constraint 4]

So you draw this and find the region involved is the triangle bounded by [1]&[2], [1]&[4], [2]&[4] (i.e. the constraint [3] is redundant).

Thoe intersections points are:

[1]&[2]: a≥b & a+b ≤ 20,000

(a,b)=(10,000 , 10,000)

[1]&[4]: a≥b & b≤8000

(a,b)=(8,000 , 8,000)

[2]&[4]: a+b ≤ 20,000 & b≤8000

(a,b)=(12,000 , 8,000)

Check each extreme point for the maximum of Return(a,b):

(a,b)=(10,000 , 10,000) : 0.1a + 0.15b = 2,500

(a,b)=(8,000 , 8,000) : 0.1a + 0.15b = 2,000

(a,b)=(12,000 , 8,000) : 0.1a + 0.15b = 2,400

Thus return is maximized at $2,500 by buying $10,000 of type A and $10,000 of type B.