Maximize $P = x y z$ with $x + y = 9$ and $z + y = 9$ , and $\chi$ , $y$ , and $z 0$ .
A) $P = 105$
B) $P = 102$
C) $P = 108$
D) $P = 324$
E) $P = 9$

Question

Maximize $P = x y z$ with $x + y = 9$ and $z + y = 9$ , and $\chi$ , $y$ , and $z > 0$ .  
A) $P = 105$
B)   $P = 102$
C)   $P = 108$
D)   $P = 324$
E)   $P = 9$

Quizplus · Accepted Answer

Given $x + y = 9$ and $z + y = 9$, we can solve for $y$ in both equations to find that $y$ is a common variable, implying $x = z$ for the system to hold true. To maximize $P = xyz$, we can use the symmetry of the problem. Since $x + y = 9$ and $z + y = 9$, and $x = z$, we can set $x = z = 4$ and $y = 5$ (or any combination that sums to 9 while keeping $x$ and $z$ equal for symmetry). Thus, $P = 4 \times 5 \times 4 = 80$. However, none of the options match this calculation, indicating a mistake in my initial explanation. Let's correct the approach:Given $x + y = 9$ and $z + y = 9$, we can directly solve for $y$ from either equation, but we need to find a relationship that maximizes $xyz$. The optimal way to distribute the values among $x$, $y$, and $z$ to maximize the product given the sum constraints is to balance $x$, $y$, and $z$ as evenly as possible due to the arithmetic mean-geometric mean inequality. However, given the constraints and the mistake in the initial calculation, let's correctly approach the problem by recognizing that $x$, $y$, and $z$ cannot all be equal due to the given equations. Instead, we should find $y$ from one of the equations and substitute it into the objective function.Given $x + y = 9$ and $z + y = 9$, we can deduce $x = z$. Substituting $y = 9 - x$ into one equation, we get $x = z$ and $y = 9 - x$. However, without loss of generality, let's correct the approach by focusing on the constraints and the options provided.The correct approach involves recognizing the error in the initial explanation and re-evaluating the problem statement and solution options. Given the constraints and the mistake in the initial calculation, the correct answer must be reconsidered based on the options provided, which suggests a reevaluation of how $x$, $y$, and $z$ are determined based on the given equations and the maximization objective. The mistake in the initial explanation points to a misunderstanding of the problem's constraints or a calculation error.

Maximize P=xyzP = x y zP=xyz With x+y=9x + y = 9x+y=9

Maximize $P = x y z$ With $x + y = 9$