Posted by shansen1@pacbell.net (66.126.109.34) on October 06, 2003 at 02:38:52:
In Reply to: graphing inequalities problem posted by on October 05, 2003 at 23:49:46:
Let x equal total acres of beans planted @ 15 acres/day
Let y equal total acres of corn planted @ 30 acres/day
Beans earn $42 profit per acre
Corn earns $35 profit per acre
Profit equation (units planted times revenue per unit)
P = x*42 + y*35
Constraint Equation #1 (units planted/units planted/unit time = time)
x/15 + y/30 <= 15 days
Multiply this equation by 30 to get rid of the fractions:
2x + y <= 450, so
y <= -2x + 450
Constraint equation #2 (total acres planted <300)
x + y <= 300
y <= 300 - x
Graph the constraint equations:
Notice that these two constraint equations yield two straight lines which intersect somewhere in the first quadrant. The first constraint equation has a y intercept at 450 and a steep slope downward of -2.
The second constraint equation has a y intercept of 300 and a more gradual slope downward of -1.
Solve the constraint equations:
Solving these two simultaneous equations by substitution gives an intersection point of {x,y} = {150,150}
Next, calculate where the steepest equation (i.e., equation 1) intersects the x axis. This is done by setting y = zero in equation 1 giving {x,y} = {0,225}
You now have a trapezoidal shaped region in the first quadrant bounded by {x,y} = {0,0}{0,300}{150,150} and {225,0} which defines the "allowable region". Any amount of beans or corn planted which lies outside this region would violate the constraint(s).
Compute the profit funtion along the boundary:
The profit function is going to be maximum at the boundary of the region somewhere (i.e at a corner somewhere), so we have to test each point separately.
{0,0} yields a profit of zero, since nothing is planted. {0,300} is the extreme point which corresponds to the situation of planting entirely corn and no beans. The profit here is P = 0*x + y*35 or $10,500, not too shabby. Now test the point {225,0} which represents the case of all beans and no corn. Because of the slow rate of planting beans, only 225 acres is able to be planted within the 15 day constraint. The profit for this case is P = 225*42 +0*35 or $9450, less than before. Now test the point {150,150}. This represents the case of half corn and half beans. The profit in this case is P=150*42+150*35 or $11550, so this is the strategy that yields maximum profit.
You might want to double check my figures here because this thing is pretty long winded, and its getting very late in the evening, but I'm quite sure the concept is right.