Kane Manufacturing has a division that produces two models of fireplace grates, x units of model A and y units of model B. To produce each model A requires 2 lb of cast iron and 7 min of labor. To produce each model B grate requires 4 lb of cast iron and 4 min of labor. The profit for each model A grate is $1.50, and the profit for each model B grate is $2.40. If 920 lb of cast iron and 1620 min of labor are available for the production of grates per day, how many grates of each model should the division produce per day in order to maximize Kane's profits P?
P = ? subject to the constraints
cast iron ?
labor ?
x ≥ 0
y ≥ 0

2 lb of cast iron is needed for model A while 4 lb of cast iron is needed for model B. 920 lb of cast iron is available for production. This can be represented by the constraint:

2x + 4y ≤ 920 (1)

7 min of labor is needed for model A while 4 min of labor is needed for model B. 1620 min of labor is available for production per day. This can be represented by the constraint:

7x + 4y ≤ 1620 (2)

Also, x ≥ 0 , y ≥ 0

Plotting the constraints using geogebra app, The points that satisfy these constraints are:

(0, 230), (231.42, 0), (0,0) and (140, 160)

The profit for each model A grate is $1.50, and the profit for each model B grate is $2.40. The profit equation is:

Profit = 1.5x + 2.4y

We are to maximize profit. Hence:

At (0,0); max Profit = 1.5(0) + 2.4(0) = 0

At (0,230); max Profit = 1.5(0) + 2.4(230) = 552

At (231.42,0); max Profit = 1.5(231.42) + 2.4(0) = 347

At (140,160); max Profit = 1.5(140) + 2.4(160) = 594

The maximum profit is at (140, 160)