Ilona Stafford manages a small firm that produces wool rugs and cotton rugs. Her total cost per day (in dollars) equals
C = 7X21 + 9X22 − 1.5X1X2
where X1 equals the number of cotton rugs produced per day and X2 equals the number of wool rugs produced per day. Because of commitments to retail stores that sell her rugs to consumers, she must produce ten rugs per day, but any mix of wool and cotton rugs is acceptable.
a. If she wants to minimize her costs (without violating her commitment to the retail stores), how many cotton rugs and wool rugs should she produce per day? (Do not use the method of Lagrangian multipliers.)
b. Does it seem reasonable that she would want to minimize cost in a situation of this sort? Why or why not?
c. Can she produce fractional numbers of rugs per day?