Question

Chapter 8, Section 8.6, Q20

Maximize production for a manufacturing process with a $24 million budget and which uses x kilograms of one raw material and y kilograms of a second raw material to make Q=3ln(x+1)+2ln(y+1) units of product. The first raw material costs $6 million per kilogram and the second costs $3 million per kilogram.

Enter the exact answers for x and y. Round your answer for Q to three decimal places.

x=?

y=?

Q=?

Answer 1

we need to maximise Q=3ln(x+1)+2ln(y+1) such that 6x+3y≤24

for critical points Q_x=0,Q_y=0
but for there exists no values of x,y such that Q_x=0,Q_y=0
so no critical points exists in domain 6x+3y≤24

on the boundary 6x+3y=24

6x+3y=24
=>3(2x+y)=24
=>2x+y=8
=>y=8-2x

Q=3ln(x+1)+2ln(y+1)
=>Q=3ln(x+1)+2ln(8-2x+1)
=>Q=3ln(x+1)+2ln(9-2x)

for crtical point ,Q'=0

=>x=2.3

=>y=3.4

so answers are

x=2.3

y=3.4

Q=6.545