A company manufactures running shoes and basketball shoes. The total revenue (in thousands of dollars) from x1 units of running shoes and x2 units of basketball shoes is R = −5x12 − 8x22 − 2x1x2 + 34x1 + 116x2, where x1 and x2 are in thousands of units. Find x1 and x2 so as to maximize the revenue.

x1 = 2000units, x2 = 7000units

Step-by-step explanation:

Given the total revenue in dollars of a company from x1 units of running shoes and x2 units of basketball shoes as;

R(x1,x2) = -5x1² − 8x2² − 2x1x2 + 34x1 + 116x2

The revenue function for running shoes will be gotten by differentiating the total revenue function with respect to x1 to have;

Rx1 = -10x1 - 2x2 + 34

Similarly to get the revenue function for units of basketball. We will have;

Rx2 = -16x2 - 2x1 + 116

Both products generates maximum/minimum revenue at their turning point i.e when Rx1 = 0 and Rx2 = 0

If Rx1 = 0 and Rx2 = 0 then;

Rx1 = -10x1 - 2x2 + 34 = 0... (1)

Rx2 = -16x2 - 2x1 + 116 ... (2)

Solving equation 1 and 2 simultaneously to get x1 and x2 at maximum revenue;

Using elimination method;

-10x1 - 2x2 = -34... (1) × 1

-16x2 - 2x1 = -116.. (2) × 5

The equations becomes;

-10x1 - 2x2 = -34

-80x2-10x1 = -580

Subtracting the resulting simultaneous equations will give;

-2x2+80x2 = -34+580

78x2 = 546

x2 = 546/78

x2 = 7

Substituting x2 = 7 into any of the equation we will have;

-10x1 - 2(7) = -34

-10x1 - 14= -34

-10x1 = -34+14

-10x1 = -20

x1 = 2

(x1,x2) = (2,7) will be our critical point

We must check if the critical point is at maximum first before we can find the maximum revenue generated for each product

Condition for the point to be maximum;

Rx1x1 < 0 and Rx1x1Rx2x2 - (Rx1x2)² >0

Rx1x1 = -10, Rx2x2 = -16, Rx1x2 = -2

It can been seen from the values that

Rx1x1 = -10<0

Also Rx1x1Rx2x2 - (Rx1x2) = -10(-16)-(-2)²

= 160-4

= 156>0

Since both conditions are met then the critical point is a maximum point. We can the calculate our maximum revenue for both products

Since x1 and x2 are in thousands of units,

x1 = 2×1000

x1 = 2000units

x2 = 7×1000

x2 = 7000units

The values of x1 and x2 needed so as to maximize the revenue is x1 = 2000units, x2 = 7000units

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step-by-step explanation:

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