Question

The variables p and q are related by the equation p^3q=9 . Another variable z, is defined by z=16p+3q . Find the value of p and of q that make z a minimum.
Just to double check my answer @rational

Answer 1

p^3q = 9

solve q from above equation and substitute in the expression you want to minimize

Answer 2

\[p^3q=9-First~eqn\]\[z=16p+3q-Second~eqn\]\[q=\frac{ 9 }{ p^3 }-Third~eqn\]\[z=16p+3(\frac{ 9 }{ p^3 })\]\[z=16p+\frac{ 27 }{ p^3 }\]\[z=16p+27(p^{-3})\]\[\frac{ dz }{ dp }=16-81(p^{-4})\]\[=16-\frac{ 81 }{ p^4 }\]\[when~\frac{ dz }{ dp }=0,\]\[16-\frac{ 81 }{ p^4 }=0\]\[16p^4=81\]\[p^4=\frac{ 81 }{ 16 }\]\[p=\frac{ 3 }{ 2 }\]

Answer 3

i'm getting p=3/2 but the answer say it is p=5/2,
i think the book's answer is wrong

Answer 4

i'm also getting q=8/3 but the answer say it is 3/8

Answer 5

is my answer correct ? @rational

Answer 6

that looks good

p = -3/2 and q = -8/3 also works right ?

Answer 7

\(p^4=\frac{ 81 }{ 16 } \implies p = \pm \dfrac{3}{2}\)

Answer 8

Thank you @rational :)

Answer 9

so the answer is \[p=\pm \frac{ 3 }{ 2 }and~q=\pm \frac{ 8 }{ 3 }\]

Answer 10

Yep!