Question

Your friend's portfolio manager has suggested two high-yielding stocks: Consolidated Edison (ED) and Royal Bank of Scotland (RBS-K). ED shares cost $40 and yield 5.5% in dividends. RBS-K shares cost $25 and yield 7.5% in dividends. You have up to $25,200 to invest, and would like to earn at least $1,386 in dividends. Draw the feasible region that shows how many shares in each company you can buy. Find the corner points of the region. (Round each coordinate to the nearest whole number.)  
Let the number of shares in ED be x, and the number of shares in RBS-K be y.
 
A)  
B)   
C)     
D)   
E)

Accepted Answer

Let x be the number of shares of ED, and y be the number of shares of RBS-K.

The total cost of buying x shares of ED and y shares of RBS-K is:

40x + 25y

The total dividends earned from x shares of ED and y shares of RBS-K is:

0.055x + 0.075y

We want to find the feasible region that shows how many shares in each company we can buy, given that we have up to $25,200 to invest and would like to earn at least $1,386 in dividends.

So we have the following two constraints:

40x + 25y ≤ 25,200 (total cost of buying the shares cannot exceed $25,200)
0.055x + 0.075y ≥ 1,386 (total dividends earned must be at least $1,386)

We also have the non-negativity constraint:

x ≥ 0 (cannot buy negative shares of ED)
y ≥ 0 (cannot buy negative shares of RBS-K)

To graph the feasible region, we first graph the two boundary lines: 40x + 25y = 25,200 and 0.055x + 0.075y = 1,386.

40x + 25y = 25,200 can be rewritten as:

y = (-40/25)x + (25,200/25) = (-8/5)x + 1008

0.055x + 0.075y = 1,386 can be rewritten as:

y = (-0.733)x + 18,480

We plot these lines and shade in the feasible region:

The corner points of the feasible region are (0, 561), (260, 216), and (630, 0).

To find the corner points, we solve for the intersection points of the two boundary lines:

(-8/5)x + 1008 = (-0.733)x + 18,480

Solving for x, we get:

x = 259.76 (rounded to 260)
y = 216.44 (rounded to 216)

So the first corner point is (260, 216).

Next, we solve for the intersection of 40x + 25y = 25,200 and y = 0:

40x + 25(0) = 25,200

Solving for x, we get:

x = 630
y = 0

So the second corner point is (630, 0).

Finally, we solve for the intersection of 0.055x + 0.075y = 1,386 and x = 0:

0.055(0) + 0.075y = 1,386

Solving for y, we get:

y = 18,480/0.075 = 246,400/3 = 82,133.33 (rounded to 82)

So the third corner point is (0, 82).

Therefore, the feasible region is a triangle with corner points (0, 82), (260, 216), and (630, 0). The maximum dividend earned is achieved at the corner point (260, 216) and is:

0.055(260) + 0.075(216) = $27.90

To earn at least $1,386 in dividends, we need:

0.055x + 0.075y ≥ 1,386

0.055x + 0.075y = 1,386

Solving for y, we get:

y = (-0.733)x + 18,480

We round x down to 259 and calculate y:

y = (-0.733)(259) + 18,480 = 183.48 (rounded to 183)

So we should buy 259 shares of ED and 183 shares of RBS-K to earn at least $1,386 in dividends. The total cost is:

40(259) + 25(183) = $18,740

And the total dividends earned is:

0.055(259) + 0.075(183) = $14.74 + $13.73 = $28.47

Note that we could also buy more shares of ED and fewer shares of RBS-K to achieve the same dividend amount, but this would increase the total cost.

Quiz 6: Linear Programming