Yawaka manufactures motorcycles and bicycles To stay in busi

Yawaka manufactures motorcycles and bicycles. To stay in business, the number of bicycles made cannot exceed two times the number of motorcycles. They lack the facilities to produce more than 90 motorcycles, or more than 140 bicycles and the total production of motorcycles and bicycles cannot exceed 210. If Yawaka makes $1270 on each motorcycle and $270 on each bicycle, how many of each should be made to maximize profit?

Yawaka should make __ motorcycles and ___ bicycles

Solution

Let the number of bicycles and motorcycles made by Yawaka be x and y respectively. Then x 140 , y 90 and x 2y. Also x + y 210. The profit on x bicycles and y motorcycles is 270x + 1270y.

Since the profit on a motorcycle is far greater than the that on a bicycle, to maximize profit, the number of motorcycles has to be maximized within the given constraints. The maximum number of motorcycles that Yawaka can produce is 90. If we assume y = 90, then since x + y   210, therefore x   210 -90 or, x   120. If x = 120 and y = 90, then the inequality x   2y is also satisfied as 120 < 2 *90. Thus, to maximize profit, Yawaka should produce 120 bicycles and 90 motorcycles.