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A computer store sells two types of computers, desktops and laptops. The supplier demands that at least 150 computers be sold a month. In order to keep profits up, the number of desktops sold must be at least twice of laptops. The store pays its sales staff a $75 commission for each desk top, and a $50 commission for each lap top. How many of each type of computers must be sold to minimize commission to its sales people? What is the minimum commission?

User Saneryee
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1 Answer

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Let's assume that the number of desktops sold is x and the number of laptops sold is y.

According to the problem, we need to sell at least 150 computers, so we have the constraint:

x + y ≥ 150

Also, the number of desktops sold must be at least twice the number of laptops sold:

x ≥ 2y

The store pays $75 commission for each desktop sold, and $50 commission for each laptop sold. Therefore, the total commission paid to sales staff is:

C = 75x + 50y

To minimize the commission, we need to find the minimum value of C subject to the above constraints.

To solve the problem, we can use the method of linear programming. First, we will graph the constraints on the coordinate plane:

y-axis

|

|

| x = 2y

| /

| /

| /

| /

| /

| /

| x + y = 150

|----------------- x-axis

Let's assume that the number of desktops sold is x and the number of laptops sold is y.

According to the problem, we need to sell at least 150 computers, so we have the constraint:

x + y ≥ 150

Also, the number of desktops sold must be at least twice the number of laptops sold:

x ≥ 2y

The store pays $75 commission for each desktop sold, and $50 commission for each laptop sold. Therefore, the total commission paid to sales staff is:

C = 75x + 50y

To minimize the commission, we need to find the minimum value of C subject to the above constraints.

To solve the problem, we can use the method of linear programming. First, we will graph the constraints on the coordinate plane:

perl

Copy code

y-axis

|

|

| x = 2y

| /

| /

| /

| /

| /

| /

| x + y = 150

|----------------- x-axis

The shaded region represents the feasible region, where all the constraints are satisfied.

Next, we need to find the corner points of the feasible region. We can do this by solving the equations of the boundary lines:

At the intersection of x = 2y and x + y = 150, we have y = 50 and x = 100. So one corner point is (100, 50).

At the intersection of x + y = 150 and the x-axis, we have y = 150 and x = 0. So another corner point is (0, 150).

At the intersection of x = 2y and the y-axis, we have y = 0 and x = 0. So the third corner point is (0, 0).

Now, we can calculate the commission at each of the corner points:

At (100, 50): C = 75(100) + 50(50) = 7750

At (0, 150): C = 75(0) + 50(150) = 7500

At (0, 0): C = 75(0) + 50(0) = 0

The minimum commission occurs at the point (0, 150) where only laptops are sold, and the commission is $7500. Therefore, the store must sell 150 laptops to minimize the commission paid to sales staff.

User Petschekr
by
8.1k points