Question

A manufacturer is producing two types of units. Each unit Q costs $9 for parts and $15 for labor and each unit R costs $6 for parts and $20 for labor. The manufacturer's budget is $810 for parts and $1800 for labor. If the income per unit is $250 for Q and $175 for R, how many units of each should be manufactured to maximize income?

Answer 1

income per unit from Q is 226 per from R is 149 slope of part cost is 6/9 and labor is 20/15 or 1.333

so Q is parts limited dQ/dR is 226/149 or i need to product 1.517 R's to make up for not producing 1 Q. from part cost I gain 1.5 R's for each Q i do not produce so a wash. From labor I gain 0.75 R's for each Q I do not produce.

So I produce all Q's to maximize income.

income Q = 250*x - 9*x(p) - 15*x(l) 9*x(p) LE 810 or x(p) LE 90 15*x(L) LE 1800 or x(L) LE 120

so the max number of Q that can be produced is 90 yielding and income of 22500-8100-1350 = 13050

As for R income is 175*y - 6*y(p) - 20*y(l) same as above x(p) LE 135 x(L) =90 so 90 is the max production and income = 15750 - 540 - 1800 =13410

Total income is 250*x + 175*y - 9*x(p) - 15*x(l) - 6*y(p) - 20*y(L) and x + y=90

250*x + 175*(90-x) - 9*x(p) - 15*x(l) - 6*(90-x) - 20*(90-x) = 77x +13410

d Income/dx = 0 = 250 +175dydx -9 -15 -20dy/dx - 6dy/dx or 226 = - 149dy/dx dy/dx = -1.517