Optimizing Chemical Z Production: $320K Budget

[andyk23]

A chemical manufacturing plant can produce z units of chemical Z given p units of chemical P and r units of chemical R, where:

z=90 p^.5 r^0.5

Chemical P costs $400 a unit and chemical R costs $3,200 a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $320,000.

First thing I did was find the constraint being 400p+3200r=320,000, then solved for p=800-8r
plugged that into the first eqn z=90(800-8r)^.5*r^.5
Then I got z=1800(2r)^.5-720r, took the derivative z'=(1800/(2r^.5))-720 set it equal to zero and solved for r. I got r=25/4
then plugged that into the eqn to solve for p, 400p+3200(25/4)=320,000, p=758.
Plugged both of those into the original eqn to find z=90(758)^.5*(25/4)^.5=6194.65...
Somewhere I'm wrong, not quite sure any guidance would help. Thanks

 

[Ray Vickson]

A chemical manufacturing plant can produce z units of chemical Z given p units of chemical P and r units of chemical R, where:

z=90 p^.5 r^0.5

Chemical P costs $400 a unit and chemical R costs $3,200 a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $320,000.

First thing I did was find the constraint being 400p+3200r=320,000, then solved for p=800-8r
plugged that into the first eqn z=90(800-8r)^.5*r^.5
Then I got z=1800(2r)^.5-720r, took the derivative z'=(1800/(2r^.5))-720 set it equal to zero and solved for r. I got r=25/4
then plugged that into the eqn to solve for p, 400p+3200(25/4)=320,000, p=758.
Plugged both of those into the original eqn to find z=90(758)^.5*(25/4)^.5=6194.65...
Somewhere I'm wrong, not quite sure any guidance would help. Thanks

Your equation $z = 90\sqrt{r} \sqrt{800-8r}$ is correct, but your equation $z = 1800 \sqrt{2r}-720r$ is incorrect. I cannot imagine how how got the second equation, or why you would think it is valid. Certainly, you don't need it: the first equation is perfectly OK for finding the optimum. Although you say you are not quite sure any guidance would help, I am offering it anyway.

RGV