A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,300, and risk factors make it prudent to invest some money in all three investments. Let x denote the amount invested at 8 %, y denote the amount invested at 7%, and z denote the amount invested at 10%. Find a general description for the amounts invested at the three rates.
A) , With
B) , With
C) , With
D) , With
E) , With

Question

A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,300, and risk factors make it prudent to invest some money in all three investments. Let x denote the amount invested at 8 %, y denote the amount invested at 7%, and z denote the amount invested at 10%. Find a general description for the amounts invested at the three rates. ​
A) , With 
B) , With 
C) , With 
D) , With 
E) , With

Quizplus · Accepted Answer

Let x denote the amount invested at 8%, y denote the amount invested at 7%, and z denote the amount invested at 10%. Then we have the system of equations:
x + y + z = 220,000 (total amount invested)
0.08x + 0.07y + 0.1z = 16,300 (interest earned)
We can solve this system using substitution or elimination. For simplicity, we will use substitution:
From the first equation, we have z = 220,000 - x - y.
Substitute this into the second equation:
0.08x + 0.07y + 0.1(220,000 - x - y) = 16,300
0.08x + 0.07y + 22,000 - 0.1x - 0.1y = 16,300
0.5x - 0.03y = -5700
Solve for y:
y = (0.5/0.03)x + 190,000
Now we can find the values of x and z using the first equation:
x + y + z = 220,000
x + (0.5/0.03)x + 190,000 + z = 220,000
(1.03/0.03)x + z = 30,000
z = 30,000 - (1.03/0.03)x
Since we want to invest some money in all three investments, we know that x, y, and z must all be positive. We also want to minimize the total amount invested, so we want to minimize x + y + z subject to the above constraints. This means we want to minimize x subject to the constraints:
y = (0.5/0.03)x + 190,000
z = 30,000 - (1.03/0.03)x
x > 0
y > 0
z > 0
We can graph this system of inequalities to find the feasible region:
Graph image: https://i.imgur.com/YxFIPeK.png
The feasible region is the triangular region bounded by the x-axis, y-axis, and the blue line. The minimum value of x occurs at the corner of this region where the blue line intersects the x-axis. This point has x = 113,500, y = 1,975,000/13, z = 105,500.
Therefore, the general description for the amounts invested at the three rates is:
x = $113,500 invested at 8%
y = $152,692.31 invested at 7%
z = $105,500 invested at 10%.

A Trust Account Manager Has $220,000 to Be Invested