Question

Goldilocks needs to find at least 12 lb of gold and at least 18 lb of silver to pay the monthly rent. There are two mines in which Goldilocks can find gold and silver. Each day that Goldilocks spends in mine 1, she finds 2 lb of gold and 2 lb of silver. Each day that Goldilocks spends in mine 2, she finds 1 lb of gold and 3 lb of silver. Formulate an LP to help Goldilocks meet her requirements while spending as little time as possible in the mines. Graphically solve the LP.

Answer 1

Answer:

To minimize the time spent to achieve the goal, Goldilocks

must spend 4 days and a half in mine 1 and 3 days in mine 2

Step-by-step explanation:

Let's call x the number of days Goldilocks spends on mine 1 and y the number of days it spends on mine 2 to meet the requirements.

Then  

2x + y is the amount of gold found in both mines and

2x + 3y  is the amount of silver found in both mines

We have the constraints  

a) 2x + y≥ 12

b) 2x + 3y ≥ 18

and we want to minimize the total time spent to accomplish this goal. That is, we want to find x and y such that

x+y is minimum.

We then draw in the xy-plane the regions a) and b) and find the feasible region which is the intersection of both regions

(SEE PICTURE ATTACHED)

The line 2x + y =12 (GREEN) has x-intersect x=6 and y-intersect y=12

The line 2x + 3y =18 (BLUE) has x-intersect x=9 and y-intersect y=6

The point P is the intersection (x,y) of the two lines, that is, the result of solving the linear system

2x + y = 12

2x + 3y = 18

The solution of this system can easily be found and gives us

P=(4.5, 3)

As the function z = x + y is linear, its minimum is attained in one of the vertices (0,12), (9,0) or (4.5,3).

A direct evaluation of z in each one this points, gives as a result that the minimum is reached in (4.5,3)

So, to minimize the time spent to achieve the goal, Goldilocks

must spend 4 days and a half in mine 1 and 3 days in mine 2.