Answer:
x1 = 2000units, x2 = 7000units
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