Linear Programming

Question

A woman seeks to invest no more than $20,000 of her savings in municipal bonds and mutual funds. The amount she will invest in municipal bonds will be at least three times the amount she will invest in mutual funds. However, she will invest no more than $5,000 in mutual funds. Under her constraints, which graph below indicates the possible ways the woman can invest her savings?

The constraints are:

(1) b > 0; f > 0 (i.e., no negative dollar values)

(2) b + f ≤ 20000 (i.e., total investment will be no more than $20,000)

(3) b ≥ 3f (i.e., the amount invested in municipal bonds will be at least three times that in mutual funds)

(4) f ≤ 5000 (i.e., no more than $5,000 invested in mutual funds)

I am unable to put the correct graph into my question, but am looking at all 4 and have no clue as to what I am trying to discern from this information. Can anyone give me a roadmap as to what I am looking for in the correct graph? It looks like a triangle with the vertex on top, if that helps. Any

Answer 1

To graph the constraints, change them to equalities

B+F = 20 (thousand)

B=3F

F=5

Put F on the y axis, B on the x axis

F=5 adds nothing. It's redundant since satisfying the other two constraints automatically satisfies F<5

B+F = 20 is a downward sloping line, at negative 45 degrees, with x intercept = 20 and y intercept =20

B+3F is an upward sloping line through the origin

those two lines intersect at (15,5) the optimal point with 5,000 invested in mutual funds and 15000 invested in bonds. The triangle with vertices (15,5) (20,0) and (0,0) enclose all the possibilities of investing

shade every thing below the B+F =20 line as the inequality allowed B+F<20

and shade everything below B=3F as the allowed inequality, B>3F

where the 2 shadings of those inequalities overlap gives the potential investments possible given the constrains

Answer 2

Equations 2 and 3 imply equation 4, so we can disregard. As b >= 3f and f >= 0, b >= 0 is also redundant.

I am assuming that f is"y" and b is "x" based upon alphabetical order.

b >= 3f implies a boundary of f = 1/3 b. Note that it goes through (0,0). As b >=3f, the feasible region is clockwise of this line. Thus, f >= 0 becomes the border as we move the clockwise. (the "x" axis).

Then, the other side of the triangle is b + f <= 20000

This meets f = 0 at b = 20000. This meet b = 3f at 3f+f = 20000 or 4f = 20000 or f = 5000, b = 3f =15000

Thus, the vertices of our feasible region are (0, 0), (20000, 0), (15000, 5000) If, in your picture, the "y" axis is b, then the order of the coordinates for the vertices of the triangle are switched.

Answer 3

The key is to look at the shading of your graph. The solution is the one where all 3 are shaded. When I graphed them, it is the area inside the triangle. Also be aware when graphing inequalities that you should be looking for the solid line, dashes are used for the < or > symbols.